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t s r t rs t s

t s rtrs t - PowerPoint PPT Presentation

Jul 14, 2023 •163 likes •1.18k views

t s rtrs t s s sts str rst s

  1. ◗✉❡st✐♊♥ ■✳ ■s t❀✐s ❝♊♥❥❡❝t✉r❡ tr✉❡ ❢♊r ❊ ✱ t❀❡ ❇❛♥❛❝❀ ❛❧❣❡❜r❛ ♊❢ ❛❧❧ ❜♊✉♥❞❡❞✱ ❧✐♥❡❛r ♊♣❡r❛t♩rs ❛❝t✐♥❣ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ❄ ❚❀❡ ❉❛❧❡s✕➏❡❧❛③❊♊ ❝♊♥❥❡❝t✉r❡ ❈♊♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❊♊ ✷✵✶✷✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❀ ❛❧❣❡❜r❛ s✉❝❀ t❀❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❀❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧✳ ❊✈✐❞❡♥❝❡✳ ❋❡rr❡✐r❛ ❛♥❞ ❚♩♠❛ss✐♥✐ ✭✶✟✌✜✮ ❀❛✈❡ ♣r♊✈❡❞ ❛ str♊♥❣❡r ❢♊r♠ ♊❢ t❀✐s ❝♊♥❥❡❝t✉r❡ ❢♊r A ❝♩♠♠✉t❛t✐✈❡✳ ◆♊♥✲❝♊♠♠✉t❛t✐✈❡ ❝❛s❡✿ ♊♣❡♥✊ ✾

  2. ❚❀❡ ❉❛❧❡s✕➏❡❧❛③❊♊ ❝♊♥❥❡❝t✉r❡ ❈♊♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❊♊ ✷✵✶✷✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❀ ❛❧❣❡❜r❛ s✉❝❀ t❀❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❀❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧✳ ❊✈✐❞❡♥❝❡✳ ❋❡rr❡✐r❛ ❛♥❞ ❚♩♠❛ss✐♥✐ ✭✶✟✌✜✮ ❀❛✈❡ ♣r♊✈❡❞ ❛ str♊♥❣❡r ❢♊r♠ ♊❢ t❀✐s ❝♊♥❥❡❝t✉r❡ ❢♊r A ❝♩♠♠✉t❛t✐✈❡✳ ◆♊♥✲❝♊♠♠✉t❛t✐✈❡ ❝❛s❡✿ ♊♣❡♥✊ ◗✉❡st✐♊♥ ■✳ ■s t❀✐s ❝♊♥❥❡❝t✉r❡ tr✉❡ ❢♊r A = B ( ❊ ) ✱ t❀❡ ❇❛♥❛❝❀ ❛❧❣❡❜r❛ ♊❢ ❛❧❧ ❜♊✉♥❞❡❞✱ ❧✐♥❡❛r ♊♣❡r❛t♩rs ❛❝t✐♥❣ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ❄ ✾

  3. ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥ ♊❢ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ❊ ♥ ♥ ♊❢ ♥♊♥✲③❡r♊✱ ❝❧♩s❡❞ s✉❜s♣❛❝❡s ♊❢ ❊ s✉❝❀ t❀❛t✱ ❢♊r ❡❛❝❀ ① ❊ ✱ t❀❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ① ♥ ✇✐t❀ ① ♥ s✉❝❀ t❀❛t ❊ ♥ ♥ ✉♥❝♊♥❞✐t✐♊♥❛❧❧②✱ ① ① ♥ ♥ ✶ t❀❛t ✐s✱ t❀❡ ♥❡t ① ♥ ◆ ✜♥✐t❡ ♥ ◆ ❝♊♥✈❡r❣❡s t♩ ① ✳ ❘❡♠❛r❊✳ ❚❀❡ ❛❜♊✈❡ t❀❡♊r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♩ ♥♊♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ✇✐t❀ ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♊♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❀❡r❡ c = ✷ ℵ ✵ ✳ ✹

  4. ❚❡r♠✐♥♊❧♊❣②✳ ❆ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥ ♊❢ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ❊ ♥ ♥ ♊❢ ♥♊♥✲③❡r♊✱ ❝❧♩s❡❞ s✉❜s♣❛❝❡s ♊❢ ❊ s✉❝❀ t❀❛t✱ ❢♊r ❡❛❝❀ ① ❊ ✱ t❀❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ① ♥ ✇✐t❀ ① ♥ s✉❝❀ t❀❛t ❊ ♥ ♥ ✉♥❝♊♥❞✐t✐♊♥❛❧❧②✱ ① ① ♥ ♥ ✶ t❀❛t ✐s✱ t❀❡ ♥❡t ① ♥ ◆ ✜♥✐t❡ ♥ ◆ ❝♊♥✈❡r❣❡s t♩ ① ✳ ❘❡♠❛r❊✳ ❚❀❡ ❛❜♊✈❡ t❀❡♊r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♩ ♥♊♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ✇✐t❀ ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♊♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❀❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ✹

  5. t❀❛t ✐s✱ t❀❡ ♥❡t ① ♥ ◆ ✜♥✐t❡ ♥ ◆ ❝♊♥✈❡r❣❡s t♩ ① ✳ ❘❡♠❛r❊✳ ❚❀❡ ❛❜♊✈❡ t❀❡♊r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♩ ♥♊♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ✇✐t❀ ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♊♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❀❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥ ♊❢ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ( ❊ ♥ ) ♥ ∈ N ♊❢ ♥♊♥✲③❡r♊✱ ❝❧♩s❡❞ s✉❜s♣❛❝❡s ♊❢ ❊ s✉❝❀ t❀❛t✱ ❢♊r ❡❛❝❀ ① ∈ ❊ ✱ t❀❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ( ① ♥ ) ✇✐t❀ ① ♥ ∈ ❊ ♥ ( ♥ ∈ N ) s✉❝❀ t❀❛t ∞ ï¿œ ✉♥❝♊♥❞✐t✐♊♥❛❧❧②✱ ① = ① ♥ ♥ = ✶ ✹

  6. ❘❡♠❛r❊✳ ❚❀❡ ❛❜♊✈❡ t❀❡♊r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♩ ♥♊♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ✇✐t❀ ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♊♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❀❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥ ♊❢ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ( ❊ ♥ ) ♥ ∈ N ♊❢ ♥♊♥✲③❡r♊✱ ❝❧♩s❡❞ s✉❜s♣❛❝❡s ♊❢ ❊ s✉❝❀ t❀❛t✱ ❢♊r ❡❛❝❀ ① ∈ ❊ ✱ t❀❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ( ① ♥ ) ✇✐t❀ ① ♥ ∈ ❊ ♥ ( ♥ ∈ N ) s✉❝❀ t❀❛t ∞ ï¿œ ✉♥❝♊♥❞✐t✐♊♥❛❧❧②✱ ① = ① ♥ ♥ = ✶ t❀❛t ✐s✱ t❀❡ ♥❡t ￜￜ ï¿œ ① ♥ ◆ ⊂ N ✜♥✐t❡ ♥ ∈ ◆ ❝♊♥✈❡r❣❡s t♩ ① ✳ ✹

  7. ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ✇✐t❀ ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♊♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❀❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥ ♊❢ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ( ❊ ♥ ) ♥ ∈ N ♊❢ ♥♊♥✲③❡r♊✱ ❝❧♩s❡❞ s✉❜s♣❛❝❡s ♊❢ ❊ s✉❝❀ t❀❛t✱ ❢♊r ❡❛❝❀ ① ∈ ❊ ✱ t❀❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ( ① ♥ ) ✇✐t❀ ① ♥ ∈ ❊ ♥ ( ♥ ∈ N ) s✉❝❀ t❀❛t ∞ ï¿œ ✉♥❝♊♥❞✐t✐♊♥❛❧❧②✱ ① = ① ♥ ♥ = ✶ t❀❛t ✐s✱ t❀❡ ♥❡t ￜￜ ï¿œ ① ♥ ◆ ⊂ N ✜♥✐t❡ ♥ ∈ ◆ ❝♊♥✈❡r❣❡s t♩ ① ✳ ❘❡♠❛r❊✳ ❚❀❡ ❛❜♊✈❡ t❀❡♊r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♩ ♥♊♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✳ ✹

  8. t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ ✵ ✶ ✶ ♣ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ✶ ♣ ❛♥❞ ❝ ✵ ♣ ❡①❛♠♣❧❡s✿ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  9. t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ ✵ ✶ ✶ ♣ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ❡①❛♠♣❧❡s✿ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ ❡①❛♠♣❧❡s✿ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ✶ ♣ ❛♥❞ ❝ ✵ ♣ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  10. t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ ✵ ✶ ✶ ♣ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  11. t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  12. t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ; ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  13. t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ; ◮ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  14. t❀❡ s♣❛❝❡ ❈ ❑ ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ; ◮ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  15. t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ; ◮ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ◮ t❀❡ s♣❛❝❡ ❈ ( ❑ ) ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  16. P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ; ◮ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ◮ t❀❡ s♣❛❝❡ ❈ ( ❑ ) ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ ◮ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ( ✶ < ♣ < ∞ ) ✳ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ ✺

  17. ❆♣♣❧✐❝❛t✐♊♥s ♊❢ t❀❡ t❀❡♊r❡♠ ■♥ ❡❛❝❀ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝❛s❡s✱ t❀❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❀❛s ❛ ❝♊✉♥t❛❜❧❡✱ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❞❡❝♊♠♣♊s✐t✐♊♥✿ ◮ ❊ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s ✭t❀✐s ♠❡❛♥s t❀❛t ❡❛❝❀ ♊❢ t❀❡ s✉❜s♣❛❝❡s ❊ ♥ ❀❛s ❞✐♠❡♥s✐♊♥ ♊♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ ï¿œ ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❀❡ ▲♊r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❀❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❀ â–Œ ; ◮ ♠♩r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♊♥t❛✐♥s ❛ ❝♊♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❀❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ◮ t❀❡ s♣❛❝❡ ❈ ( ❑ ) ♊❢ ❝♊♥t✐♥✉♊✉s ❢✉♥❝t✐♊♥s ♊♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ ◮ t❀❡ ♣ t❀ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ( ✶ < ♣ < ∞ ) ✳ ❍❡♥❝❡ ♥♊t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❀ ♊❢ t❀❡ ❛❜♊✈❡ ❝❛s❡s✳ P❡rs♣❡❝t✐✈❡✳ ●♊✇❡rs✬s ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡ ✐s ❡✐t❀❡r ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ♥♊♥❡ ♊❢ ✐ts ❝❧♩s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♊♠♣♊s❡❞ ✐♥t♩ t❀❡ ❞✐r❡❝t s✉♠ ♊❢ t✇♩ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡s✮✱ ♩r ✐t ❝♊♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❀✐❝❀ ❀❛s ❛♥ ✉♥❝♊♥❞✐t✐♊♥❛❧ ❙❝❀❛✉❞❡r ❜❛s✐s✳ ✺

  18. ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ ①✳ ▌♊r❡♊✈❡r✱ ✵ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ① ② ❊ ① ② ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ① ❢♊r s♊♠❡ ① ✵ ✐s ❊ ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❊❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✻

  19. ▌♊r❡♊✈❡r✱ ✵ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ① ② ❊ ① ② ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ① ❢♊r s♊♠❡ ① ✵ ✐s ❊ ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❊❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ✻

  20. ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ① ❢♊r s♊♠❡ ① ✵ ✐s ❊ ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❊❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ✻

  21. ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❊❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ML ① ❢♊r s♊♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ✻

  22. ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❊❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ML ① ❢♊r s♊♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ✻

  23. Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❊❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ML ① ❢♊r s♊♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ✻

  24. ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❝♩rr❡s♣♊♥❞s t♩ ❊ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ML ① ❢♊r s♊♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❋ ￜ→ { ❚ ∈ B ( ❊ ) : ❋ ⊆ ❊❡r ❚ } ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✳ ✻

  25. ❛♥❞ t❀❡r❡❢♊r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ML ① ❢♊r s♊♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❋ ￜ→ { ❚ ∈ B ( ❊ ) : ❋ ⊆ ❊❡r ❚ } ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ❝♩rr❡s♣♊♥❞s t♩ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ✻

  26. ❆ r❡✜♥❡♠❡♥t ♊❢ t❀❡ q✉❡st✐♊♥ ❖❜s❡r✈❛t✐♊♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡✳ ❋♩r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ❣❡♥❡r❛t❡❞ ❜② t❀❡ s✐♥❣❧❡ ♣r♊❥❡❝t✐♊♥ ■ − P✱ ✇❀❡r❡ P ✐s ❛♥② ♣r♊❥❡❝t✐♊♥ ♊❢ ❊ ♊♥t♩ C ①✳ ▌♊r❡♊✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♊♣♊rt✐♊♥❛❧ ❚❡r♠✐♥♊❧♊❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ t❀❡ ❢♊r♠ ML ① ❢♊r s♊♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♊♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❀✐s ✐s tr✉❡ ❢♊r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ Pr♊♊❢✳ ❚❀❡ ♠❛♣♣✐♥❣ ❋ ￜ→ { ❚ ∈ B ( ❊ ) : ❋ ⊆ ❊❡r ❚ } ✐s ❛♥ ❛♥t✐✲✐s♩♠♩r♣❀✐s♠ ♊❢ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♊❢ ❊ ♊♥t♩ t❀❡ ❧❛tt✐❝❡ ♊❢ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ❝♩rr❡s♣♊♥❞s t♩ ❛ ✉♥✐q✉❡ ♊♥❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ✱ ❛♥❞ t❀❡r❡❢♊r❡ L ✐s ✜①❡❞✳ ✷ ✻

  27. ❍❡♥❝❡ ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❊ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✳ ✵ ✳ ❋❛❝t✳ ❊ ① ❢♊r ❡❛❝❀ ① ❊ Pr♊♊❢✳ ●✐✈❡♥ ① ✵ ✱ t❛❊❡ s✉❝❀ t❀❛t ① ✶✳ ❚❀❡♥ ❊ ❊ ❚ ② ② ① ❊ ❊ ❜❡❧♊♥❣s t♩ ❊ ✱ ❜✉t ♥♊t t♩ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ✌

  28. ✵ ✳ ❋❛❝t✳ ❊ ① ❢♊r ❡❛❝❀ ① ❊ Pr♊♊❢✳ ●✐✈❡♥ ① ✵ ✱ t❛❊❡ s✉❝❀ t❀❛t ① ✶✳ ❚❀❡♥ ❊ ❊ ❚ ② ② ① ❊ ❊ ❜❡❧♊♥❣s t♩ ❊ ✱ ❜✉t ♥♊t t♩ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ✌

  29. Pr♊♊❢✳ ●✐✈❡♥ ① ✵ ✱ t❛❊❡ s✉❝❀ t❀❛t ① ✶✳ ❚❀❡♥ ❊ ❊ ❚ ② ② ① ❊ ❊ ❜❡❧♊♥❣s t♩ ❊ ✱ ❜✉t ♥♊t t♩ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ ✌

  30. ❚❀❡♥ ❚ ② ② ① ❊ ❊ ❜❡❧♊♥❣s t♩ ❊ ✱ ❜✉t ♥♊t t♩ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ Pr♊♊❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❊❡ λ ∈ ❊ ∗ s✉❝❀ t❀❛t ï¿œ ① , λ ï¿œ = ✶✳ ✌

  31. ❜✉t ♥♊t t♩ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ Pr♊♊❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❊❡ λ ∈ ❊ ∗ s✉❝❀ t❀❛t ï¿œ ① , λ ï¿œ = ✶✳ ❚❀❡♥ ❚ : ② ￜ→ ï¿œ ② , λ ï¿œ ① , ❊ → ❊ , ❜❡❧♊♥❣s t♩ F ( ❊ ) ✱ ✌

  32. ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ Pr♊♊❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❊❡ λ ∈ ❊ ∗ s✉❝❀ t❀❛t ï¿œ ① , λ ï¿œ = ✶✳ ❚❀❡♥ ❚ : ② ￜ→ ï¿œ ② , λ ï¿œ ① , ❊ → ❊ , ❜❡❧♊♥❣s t♩ F ( ❊ ) ✱ ❜✉t ♥♊t t♩ ML ① ❜❡❝❛✉s❡ ❚① = ① ï¿œ = ✵✳ ✷ ✌

  33. ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞✳ ❚❀❡♥ ❊ ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ Pr♊♊❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❊❡ λ ∈ ❊ ∗ s✉❝❀ t❀❛t ï¿œ ① , λ ï¿œ = ✶✳ ❚❀❡♥ ❚ : ② ￜ→ ï¿œ ② , λ ï¿œ ① , ❊ → ❊ , ❜❡❧♊♥❣s t♩ F ( ❊ ) ✱ ❜✉t ♥♊t t♩ ML ① ❜❡❝❛✉s❡ ❚① = ① ï¿œ = ✵✳ ✷ ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ✌

  34. ◗✉❡st✐♊♥ ■■■✳ ■s ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♊❢ ❊ ❄ ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ Pr♊♊❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❊❡ λ ∈ ❊ ∗ s✉❝❀ t❀❛t ï¿œ ① , λ ï¿œ = ✶✳ ❚❀❡♥ ❚ : ② ￜ→ ï¿œ ② , λ ï¿œ ① , ❊ → ❊ , ❜❡❧♊♥❣s t♩ F ( ❊ ) ✱ ❜✉t ♥♊t t♩ ML ① ❜❡❝❛✉s❡ ❚① = ① ï¿œ = ✵✳ ✷ ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ✌

  35. ◗✉❡st✐♊♥ ■■ ✖ t❀❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♊♣❡r✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) ￜ⊆ ML ① ❢♊r ❡❛❝❀ ① ∈ ❊ \ { ✵ } ✳ Pr♊♊❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❊❡ λ ∈ ❊ ∗ s✉❝❀ t❀❛t ï¿œ ① , λ ï¿œ = ✶✳ ❚❀❡♥ ❚ : ② ￜ→ ï¿œ ② , λ ï¿œ ① , ❊ → ❊ , ❜❡❧♊♥❣s t♩ F ( ❊ ) ✱ ❜✉t ♥♊t t♩ ML ① ❜❡❝❛✉s❡ ❚① = ① ï¿œ = ✵✳ ✷ ❈♩r♩❧❧❛r②✳ ❆ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■ ✐♠♣❧✐❡s ❛ ♣♊s✐t✐✈❡ ❛♥s✇❡r t♩ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ ❇❛♥❛❝❀ s♣❛❝❡✱ ❛♥❞ s✉♣♣♊s❡ t❀❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ❚❀❡♥ B ( ❊ ) ❝♊♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❀✐❝❀ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♊♥ ■■■✳ ■s F ( ❊ ) ❡✈❡r ❝♊♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ❄ ✌

  36. ✭✐✐✮ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡ s❡t ♊❢ ✐♥❡ss❡♥t✐❛❧ ♊♣❡r❛t♩rs ♊♥ ❊ ✐s ❛ ❝❧♩s❡❞✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ❊ ♊❢ ❊ ✱ ❛♥❞ ♣r♊♣❡r ✇❀❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♩r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♊♥ ✐s t❀❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ❜❡❧♊♥❣s t♩ t❀❡ ❏❛❝♩❜s♊♥ r❛❞✐❝❛❧ ♊❢ t❀❡ ❈❛❧❊✐♥ ❛❧❣❡❜r❛ ❊ ✳ ❚ ❊ ❊ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♊♣❡r❛t♩r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐t ✐s ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♩r ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❚ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❀♊❧♠ ♊♣❡r❛t♩r✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ❞✐♠ ❊❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♊r ❡❛❝❀ ❙ ∈ B ( ❊ ) ✳ ✜

  37. ✭✐✐✮ ❝♊♥t❛✐♥s ❊ ✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♩r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♊♥ ✐s t❀❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ❜❡❧♊♥❣s t♩ t❀❡ ❏❛❝♩❜s♊♥ r❛❞✐❝❛❧ ♊❢ t❀❡ ❈❛❧❊✐♥ ❛❧❣❡❜r❛ ❊ ✳ ❚ ❊ ❊ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♊♣❡r❛t♩r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐t ✐s ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♩r ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❚ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❀♊❧♠ ♊♣❡r❛t♩r✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ❞✐♠ ❊❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♊r ❡❛❝❀ ❙ ∈ B ( ❊ ) ✳ ❚❀❡ s❡t E ( ❊ ) ♊❢ ✐♥❡ss❡♥t✐❛❧ ♊♣❡r❛t♩rs ♊♥ ❊ ✐s ❛ ❝❧♩s❡❞✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ❛♥❞ ♣r♊♣❡r ✇❀❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ✜

  38. ✭✐✐✮ ❝♊♥t❛✐♥s ❊ ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♊♣❡r❛t♩r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐t ✐s ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♩r ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❚ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❀♊❧♠ ♊♣❡r❛t♩r✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ❞✐♠ ❊❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♊r ❡❛❝❀ ❙ ∈ B ( ❊ ) ✳ ❚❀❡ s❡t E ( ❊ ) ♊❢ ✐♥❡ss❡♥t✐❛❧ ♊♣❡r❛t♩rs ♊♥ ❊ ✐s ❛ ❝❧♩s❡❞✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ❛♥❞ ♣r♊♣❡r ✇❀❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♩r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♊♥ ✐s t❀❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♊♥❣s t♩ t❀❡ ❏❛❝♩❜s♊♥ r❛❞✐❝❛❧ ♊❢ t❀❡ ❈❛❧❊✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ✜

  39. ✭✐✐✮ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♩r ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❚ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❀♊❧♠ ♊♣❡r❛t♩r✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ❞✐♠ ❊❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♊r ❡❛❝❀ ❙ ∈ B ( ❊ ) ✳ ❚❀❡ s❡t E ( ❊ ) ♊❢ ✐♥❡ss❡♥t✐❛❧ ♊♣❡r❛t♩rs ♊♥ ❊ ✐s ❛ ❝❧♩s❡❞✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ❛♥❞ ♣r♊♣❡r ✇❀❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♩r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♊♥ ✐s t❀❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♊♥❣s t♩ t❀❡ ❏❛❝♩❜s♊♥ r❛❞✐❝❛❧ ♊❢ t❀❡ ❈❛❧❊✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♊♣❡r❛t♩r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐t ✐s ❝♊♠♣❛❝t✳ ✜

  40. ✭✐✐✮ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❚ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❀♊❧♠ ♊♣❡r❛t♩r✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ❞✐♠ ❊❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♊r ❡❛❝❀ ❙ ∈ B ( ❊ ) ✳ ❚❀❡ s❡t E ( ❊ ) ♊❢ ✐♥❡ss❡♥t✐❛❧ ♊♣❡r❛t♩rs ♊♥ ❊ ✐s ❛ ❝❧♩s❡❞✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ❛♥❞ ♣r♊♣❡r ✇❀❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♩r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♊♥ ✐s t❀❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♊♥❣s t♩ t❀❡ ❏❛❝♩❜s♊♥ r❛❞✐❝❛❧ ♊❢ t❀❡ ❈❛❧❊✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♊♣❡r❛t♩r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐t ✐s ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✜

  41. ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❚ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❀♊❧♠ ♊♣❡r❛t♩r✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ❞✐♠ ❊❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♊r ❡❛❝❀ ❙ ∈ B ( ❊ ) ✳ ❚❀❡ s❡t E ( ❊ ) ♊❢ ✐♥❡ss❡♥t✐❛❧ ♊♣❡r❛t♩rs ♊♥ ❊ ✐s ❛ ❝❧♩s❡❞✱ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ❛♥❞ ♣r♊♣❡r ✇❀❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♩r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♊♥ ✐s t❀❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♊♥❣s t♩ t❀❡ ❏❛❝♩❜s♊♥ r❛❞✐❝❛❧ ♊❢ t❀❡ ❈❛❧❊✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♊♣❡r❛t♩r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐t ✐s ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✭✐✐✮ L ❝♊♥t❛✐♥s E ( ❊ ) ✳ ✜

  42. ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❀❡ ❛♥❛❧♊❣✉❡ ♊❢ t❀❡ ❢❛❝t t❀❛t ❛♥ ✉❧tr❛✜❧t❡r ♊♥ ❛ s❡t â–Œ ✐s ❡✐t❀❡r ✜①❡❞ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ❝♊♥s✐sts ♊❢ ♣r❡❝✐s❡❧② t❀♊s❡ s✉❜s❡ts ♊❢ â–Œ ✇❀✐❝❀ ❝♊♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① â–Œ ✮✱ ♩r ✐t ❝♊♥t❛✐♥s t❀❡ ❋ré❝❀❡t ✜❧t❡r ♊❢ ❛❧❧ ❝♊✜♥✐t❡ s✉❜s❡ts ♊❢ â–Œ ✳ ❘❡♠❛r❊s✳ ❚❀✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❊♥♊✇♥ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❀❡ ❛♣♣r♊❛❝❀ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♩♠ ♩✉rs✳ ❈♩r♩❧❧❛r②✳ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❀❡ ❢♊❧❧♊✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ♥♊ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡ ❡q✉✐✈❛❧❡♥❝❡ ♊❢ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✭✐✐✮ L ❝♊♥t❛✐♥s E ( ❊ ) ✳ ✟

  43. ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❀❡ ❛♥❛❧♊❣✉❡ ♊❢ t❀❡ ❢❛❝t t❀❛t ❛♥ ✉❧tr❛✜❧t❡r ♊♥ ❛ s❡t â–Œ ✐s ❡✐t❀❡r ✜①❡❞ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ❝♊♥s✐sts ♊❢ ♣r❡❝✐s❡❧② t❀♊s❡ s✉❜s❡ts ♊❢ â–Œ ✇❀✐❝❀ ❝♊♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① â–Œ ✮✱ ♩r ✐t ❝♊♥t❛✐♥s t❀❡ ❋ré❝❀❡t ✜❧t❡r ♊❢ ❛❧❧ ❝♊✜♥✐t❡ s✉❜s❡ts ♊❢ â–Œ ✳ ❈♩r♩❧❧❛r②✳ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❀❡ ❢♊❧❧♊✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ♥♊ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡ ❡q✉✐✈❛❧❡♥❝❡ ♊❢ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✭✐✐✮ L ❝♊♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❊s✳ ◮ ❚❀✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❊♥♊✇♥ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❀❡ ❛♣♣r♊❛❝❀ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♩♠ ♩✉rs✳ ✟

  44. ❈♩r♩❧❧❛r②✳ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❀❡ ❢♊❧❧♊✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ♥♊ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡ ❡q✉✐✈❛❧❡♥❝❡ ♊❢ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✭✐✐✮ L ❝♊♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❊s✳ ◮ ❚❀✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❊♥♊✇♥ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❀❡ ❛♣♣r♊❛❝❀ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♩♠ ♩✉rs✳ ◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❀❡ ❛♥❛❧♊❣✉❡ ♊❢ t❀❡ ❢❛❝t t❀❛t ❛♥ ✉❧tr❛✜❧t❡r ♊♥ ❛ s❡t â–Œ ✐s ❡✐t❀❡r ✜①❡❞ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ❝♊♥s✐sts ♊❢ ♣r❡❝✐s❡❧② t❀♊s❡ s✉❜s❡ts ♊❢ â–Œ ✇❀✐❝❀ ❝♊♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ â–Œ ✮✱ ♩r ✐t ❝♊♥t❛✐♥s t❀❡ ❋ré❝❀❡t ✜❧t❡r ♊❢ ❛❧❧ ❝♊✜♥✐t❡ s✉❜s❡ts ♊❢ â–Œ ✳ ✟

  45. ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ♥♊ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ❝♊♥t❛✐♥s ❊ ✳ ❚❀❡ ❡q✉✐✈❛❧❡♥❝❡ ♊❢ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✭✐✐✮ L ❝♊♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❊s✳ ◮ ❚❀✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❊♥♊✇♥ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❀❡ ❛♣♣r♊❛❝❀ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♩♠ ♩✉rs✳ ◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❀❡ ❛♥❛❧♊❣✉❡ ♊❢ t❀❡ ❢❛❝t t❀❛t ❛♥ ✉❧tr❛✜❧t❡r ♊♥ ❛ s❡t â–Œ ✐s ❡✐t❀❡r ✜①❡❞ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ❝♊♥s✐sts ♊❢ ♣r❡❝✐s❡❧② t❀♊s❡ s✉❜s❡ts ♊❢ â–Œ ✇❀✐❝❀ ❝♊♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ â–Œ ✮✱ ♩r ✐t ❝♊♥t❛✐♥s t❀❡ ❋ré❝❀❡t ✜❧t❡r ♊❢ ❛❧❧ ❝♊✜♥✐t❡ s✉❜s❡ts ♊❢ â–Œ ✳ ❈♩r♩❧❧❛r②✳ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❀❡ ❢♊❧❧♊✇✐♥❣ s❡♥s❡✿ ✟

  46. ❚❀❡ ❡q✉✐✈❛❧❡♥❝❡ ♊❢ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♊♥✲③❡r♩ ❇❛♥❛❝❀ s♣❛❝❡✳ ❚❀❡♥✱ ❢♊r ❡❛❝❀ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♊❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ t✇♩ ❛❧t❡r♥❛t✐✈❡s ❀♊❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♩r ✭✐✐✮ L ❝♊♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❊s✳ ◮ ❚❀✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❊♥♊✇♥ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❀❡ ❛♣♣r♊❛❝❀ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♩♠ ♩✉rs✳ ◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❀❡ ❛♥❛❧♊❣✉❡ ♊❢ t❀❡ ❢❛❝t t❀❛t ❛♥ ✉❧tr❛✜❧t❡r ♊♥ ❛ s❡t â–Œ ✐s ❡✐t❀❡r ✜①❡❞ ✭✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ❝♊♥s✐sts ♊❢ ♣r❡❝✐s❡❧② t❀♊s❡ s✉❜s❡ts ♊❢ â–Œ ✇❀✐❝❀ ❝♊♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ â–Œ ✮✱ ♩r ✐t ❝♊♥t❛✐♥s t❀❡ ❋ré❝❀❡t ✜❧t❡r ♊❢ ❛❧❧ ❝♊✜♥✐t❡ s✉❜s❡ts ♊❢ â–Œ ✳ ❈♩r♩❧❧❛r②✳ ◗✉❡st✐♊♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❀❡ ❢♊❧❧♊✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ♥♊ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ❝♊♥t❛✐♥s F ( ❊ ) ✳ ✟

  47. ✭ ❡①❛♠♣❧❡s✿ ♣ ❛♥❞ ▲ ♣ ✵ ✶ ❢♊r ✶ ✮❀ ♣ ✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ✱ ❊ ❍✱ ♩r ❊ ❝ ✵ ❍✱ ✇❀❡r❡ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  48. ✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ✱ ❊ ❍✱ ♩r ❊ ❝ ✵ ❍✱ ✇❀❡r❡ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭ ❡①❛♠♣❧❡s✿ ♣ ❛♥❞ ▲ ♣ ✵ ✶ ❢♊r ✶ ✮❀ ♣ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  49. ✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ✱ ❊ ❍✱ ♩r ❊ ❝ ✵ ❍✱ ✇❀❡r❡ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  50. ✱ ❊ ❍✱ ♩r ❊ ❝ ✵ ❍✱ ✇❀❡r❡ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  51. ✱ ❊ ❍✱ ♩r ❊ ❝ ✵ ❍✱ ✇❀❡r❡ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  52. ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✱ ❊ ❍✱ ♩r ❊ ❝ ✵ ❍ ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  53. ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✱ ♩r ❊ ❝ ✵ ❍ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍ ✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  54. ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♩r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  55. ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ✳ ❙① P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♩r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵

  56. ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♩r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ε > ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ï¿œ ❙① ï¿œ ï¿œ ε ✳ ✶✵

  57. ✭✈✮ ❊ ❈ ❑ ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ❑ ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♩r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ε > ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ï¿œ ❙① ï¿œ ï¿œ ε ✳ ✶✵

  58. P♩s✐t✐✈❡ ❛♥s✇❡rs t♩ ◗✉❡st✐♊♥ ■■ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❀ s♣❛❝❡ ✇❀✐❝❀ s❛t✐s✜❡s ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ✜✈❡ ❝♊♥❞✐t✐♊♥s✿ ✭✐✮ ❊ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♊r ✶ ï¿œ ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❊ ✐s ❛✉t♩♠❛t✐❝❛❧❧② ❝♊♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♊r s♊♠❡ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♩r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❀❡r❡ Γ ✐s ❛ ♥♊♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❀❛s ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❀❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♊♠♣♊s❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡s✮❀ ✭✈✮ ❊ = ❈ ( ❑ ) ✱ ✇❀❡r❡ ❑ ✐s ❛ ❝♊♠♣❛❝t ❍❛✉s❞♩r✛ s♣❛❝❡ ✇✐t❀♊✉t ✐s♩❧❛t❡❞ ♣♊✐♥ts✱ ❛♥❞ ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❈ ( ❑ ) ✐s ❛ ✇❡❛❊ ♠✉❧t✐♣❧✐❝❛t✐♊♥✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♊♥ ♊♣❡r❛t♩r✳ ❚❀❡♥ ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♊♥✳ ❆♥ ♊♣❡r❛t♩r ❙ ♊♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♊r ❡❛❝❀ ε > ✵✱ ❡❛❝❀ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ♊❢ ❊ ❝♊♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♩r ① s✉❝❀ t❀❛t ï¿œ ❙① ï¿œ ï¿œ ε ✳ ✶✵

  59. ❆ st❛t❡ ♊♥ ✐s ❛ ♥♊r♠✲♊♥❡ ❢✉♥❝t✐♊♥❛❧ ♊♥ s✉❝❀ t❀❛t ✵ ❛ ❛ ❛ ●✐✈❡♥ ❛ st❛t❡ ♊♥ ✱ t❀❡ s❡t ✵ ❛ ❛ ❛ ✐s ❛ ❝❧♩s❡❞ ❧❡❢t ✐❞❡❛❧ ♊❢ ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❀❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♊✐♥t ♊❢ t❀❡ ✇❡❛❊ ✲❝♊♠♣❛❝t✱ ❝♊♥✈❡① s❡t ♊❢ ❛❧❧ st❛t❡s ♊♥ ✳ ❙✉♣♣♊s❡ t❀❛t ❢♊r s♊♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♩r✳ ❚❀❡♥ ① ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s t❀❡ ✈❡❝t♩r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❀❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❀❡r❡ ❞❡♥♊t❡s t❀❡ ✐♥♥❡r ♣r♩❞✉❝t ♊♥ ❍ ✳ ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s ❢♊❧❧♊✇s ❢r♩♠ t❀❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❀ ♣✉r❡ st❛t❡ ♊♥ ✐s ❡✐t❀❡r ❛ ✈❡❝t♩r st❛t❡✱ ♩r ❊❡r ✱ ✐♥ ✇❀✐❝❀ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❀❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❀ ✐♥✈♊❧✉t✐♊♥ ❛ ￜ→ ❛ ⋆ ✳ ✶✶

  60. ●✐✈❡♥ ❛ st❛t❡ ♊♥ ✱ t❀❡ s❡t ✵ ❛ ❛ ❛ ✐s ❛ ❝❧♩s❡❞ ❧❡❢t ✐❞❡❛❧ ♊❢ ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❀❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♊✐♥t ♊❢ t❀❡ ✇❡❛❊ ✲❝♊♠♣❛❝t✱ ❝♊♥✈❡① s❡t ♊❢ ❛❧❧ st❛t❡s ♊♥ ✳ ❙✉♣♣♊s❡ t❀❛t ❢♊r s♊♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♩r✳ ❚❀❡♥ ① ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s t❀❡ ✈❡❝t♩r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❀❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❀❡r❡ ❞❡♥♊t❡s t❀❡ ✐♥♥❡r ♣r♩❞✉❝t ♊♥ ❍ ✳ ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s ❢♊❧❧♊✇s ❢r♩♠ t❀❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❀ ♣✉r❡ st❛t❡ ♊♥ ✐s ❡✐t❀❡r ❛ ✈❡❝t♩r st❛t❡✱ ♩r ❊❡r ✱ ✐♥ ✇❀✐❝❀ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❀❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❀ ✐♥✈♊❧✉t✐♊♥ ❛ ￜ→ ❛ ⋆ ✳ ❆ st❛t❡ ♊♥ A ✐s ❛ ♥♊r♠✲♊♥❡ ❢✉♥❝t✐♊♥❛❧ λ ♊♥ A s✉❝❀ t❀❛t ï¿œ ❛ ⋆ ❛ , λ ï¿œ ï¿œ ✵ ( ❛ ∈ A ) . ✶✶

  61. ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❀❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♊✐♥t ♊❢ t❀❡ ✇❡❛❊ ✲❝♊♠♣❛❝t✱ ❝♊♥✈❡① s❡t ♊❢ ❛❧❧ st❛t❡s ♊♥ ✳ ❙✉♣♣♊s❡ t❀❛t ❢♊r s♊♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♩r✳ ❚❀❡♥ ① ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s t❀❡ ✈❡❝t♩r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❀❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❀❡r❡ ❞❡♥♊t❡s t❀❡ ✐♥♥❡r ♣r♩❞✉❝t ♊♥ ❍ ✳ ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s ❢♊❧❧♊✇s ❢r♩♠ t❀❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❀ ♣✉r❡ st❛t❡ ♊♥ ✐s ❡✐t❀❡r ❛ ✈❡❝t♩r st❛t❡✱ ♩r ❊❡r ✱ ✐♥ ✇❀✐❝❀ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❀❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❀ ✐♥✈♊❧✉t✐♊♥ ❛ ￜ→ ❛ ⋆ ✳ ❆ st❛t❡ ♊♥ A ✐s ❛ ♥♊r♠✲♊♥❡ ❢✉♥❝t✐♊♥❛❧ λ ♊♥ A s✉❝❀ t❀❛t ï¿œ ❛ ⋆ ❛ , λ ï¿œ ï¿œ ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♊♥ A ✱ t❀❡ s❡t N λ = { ❛ ∈ A : ï¿œ ❛ ⋆ ❛ , λ ï¿œ = ✵ } ✐s ❛ ❝❧♩s❡❞ ❧❡❢t ✐❞❡❛❧ ♊❢ A ✳ ✶✶

  62. ❙✉♣♣♊s❡ t❀❛t ❢♊r s♊♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♩r✳ ❚❀❡♥ ① ✐❢ ❛♥❞ ♊♥❧② ✐❢ ✐s t❀❡ ✈❡❝t♩r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❀❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❀❡r❡ ❞❡♥♊t❡s t❀❡ ✐♥♥❡r ♣r♩❞✉❝t ♊♥ ❍ ✳ ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s ❢♊❧❧♊✇s ❢r♩♠ t❀❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❀ ♣✉r❡ st❛t❡ ♊♥ ✐s ❡✐t❀❡r ❛ ✈❡❝t♩r st❛t❡✱ ♩r ❊❡r ✱ ✐♥ ✇❀✐❝❀ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❀❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❀ ✐♥✈♊❧✉t✐♊♥ ❛ ￜ→ ❛ ⋆ ✳ ❆ st❛t❡ ♊♥ A ✐s ❛ ♥♊r♠✲♊♥❡ ❢✉♥❝t✐♊♥❛❧ λ ♊♥ A s✉❝❀ t❀❛t ï¿œ ❛ ⋆ ❛ , λ ï¿œ ï¿œ ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♊♥ A ✱ t❀❡ s❡t N λ = { ❛ ∈ A : ï¿œ ❛ ⋆ ❛ , λ ï¿œ = ✵ } ✐s ❛ ❝❧♩s❡❞ ❧❡❢t ✐❞❡❛❧ ♊❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❀❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♊✐♥t ♊❢ t❀❡ ✇❡❛❊ ∗ ✲❝♊♠♣❛❝t✱ ❝♊♥✈❡① s❡t ♊❢ ❛❧❧ st❛t❡s ♊♥ A ✳ ✶✶

  63. ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s ❢♊❧❧♊✇s ❢r♩♠ t❀❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❀ ♣✉r❡ st❛t❡ ♊♥ ✐s ❡✐t❀❡r ❛ ✈❡❝t♩r st❛t❡✱ ♩r ❊❡r ✱ ✐♥ ✇❀✐❝❀ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❀❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❀ ✐♥✈♊❧✉t✐♊♥ ❛ ￜ→ ❛ ⋆ ✳ ❆ st❛t❡ ♊♥ A ✐s ❛ ♥♊r♠✲♊♥❡ ❢✉♥❝t✐♊♥❛❧ λ ♊♥ A s✉❝❀ t❀❛t ï¿œ ❛ ⋆ ❛ , λ ï¿œ ï¿œ ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♊♥ A ✱ t❀❡ s❡t N λ = { ❛ ∈ A : ï¿œ ❛ ⋆ ❛ , λ ï¿œ = ✵ } ✐s ❛ ❝❧♩s❡❞ ❧❡❢t ✐❞❡❛❧ ♊❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❀❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♊✐♥t ♊❢ t❀❡ ✇❡❛❊ ∗ ✲❝♊♠♣❛❝t✱ ❝♊♥✈❡① s❡t ♊❢ ❛❧❧ st❛t❡s ♊♥ A ✳ ❙✉♣♣♊s❡ t❀❛t A = B ( ❍ ) ❢♊r s♊♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ∈ ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♩r✳ ❚❀❡♥ N λ = ML ① ✐❢ ❛♥❞ ♊♥❧② ✐❢ λ ✐s t❀❡ ✈❡❝t♩r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❀❛t ✐s✱ ï¿œ ❚ , λ ï¿œ = ( ❚① | ① ) ( ❚ ∈ B ( ❍ )) , ✇❀❡r❡ ( · | · ) ❞❡♥♊t❡s t❀❡ ✐♥♥❡r ♣r♩❞✉❝t ♊♥ ❍ ✳ ✶✶

  64. ❚❀❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❀ ✐♥✈♊❧✉t✐♊♥ ❛ ￜ→ ❛ ⋆ ✳ ❆ st❛t❡ ♊♥ A ✐s ❛ ♥♊r♠✲♊♥❡ ❢✉♥❝t✐♊♥❛❧ λ ♊♥ A s✉❝❀ t❀❛t ï¿œ ❛ ⋆ ❛ , λ ï¿œ ï¿œ ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♊♥ A ✱ t❀❡ s❡t N λ = { ❛ ∈ A : ï¿œ ❛ ⋆ ❛ , λ ï¿œ = ✵ } ✐s ❛ ❝❧♩s❡❞ ❧❡❢t ✐❞❡❛❧ ♊❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❀❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♊✐♥t ♊❢ t❀❡ ✇❡❛❊ ∗ ✲❝♊♠♣❛❝t✱ ❝♊♥✈❡① s❡t ♊❢ ❛❧❧ st❛t❡s ♊♥ A ✳ ❙✉♣♣♊s❡ t❀❛t A = B ( ❍ ) ❢♊r s♊♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ∈ ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♩r✳ ❚❀❡♥ N λ = ML ① ✐❢ ❛♥❞ ♊♥❧② ✐❢ λ ✐s t❀❡ ✈❡❝t♩r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❀❛t ✐s✱ ï¿œ ❚ , λ ï¿œ = ( ❚① | ① ) ( ❚ ∈ B ( ❍ )) , ✇❀❡r❡ ( · | · ) ❞❡♥♊t❡s t❀❡ ✐♥♥❡r ♣r♩❞✉❝t ♊♥ ❍ ✳ ❚❀❡ ❉✐❝❀♊t♊♠② ❚❀❡♊r❡♠ ❢♊r ❍✐❧❜❡rt s♣❛❝❡s ❢♊❧❧♊✇s ❢r♩♠ t❀❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❀ ♣✉r❡ st❛t❡ λ ♊♥ B ( ❍ ) ✐s ❡✐t❀❡r ❛ ✈❡❝t♩r st❛t❡✱ ♩r K ( ❍ ) ⊆ ❊❡r λ ✱ ✐♥ ✇❀✐❝❀ ❝❛s❡ K ( ❍ ) ⊆ N λ ✳ ✶✶

  65. ✭✐✐✮ ❳ ❆❍ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❳ ❆❍ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ ❆r❣②r♩s✕❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❚❀❡♊r❡♠ ✭❆r❣②r♩s✕❍❛②❞♊♥ ✷✵✶✶✮✳ ❚❀❡r❡ ✐s ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ✇❀✐❝❀ ❀❛s t❀❡ ❢♊❧❧♊✇✐♥❣ t❀r❡❡ ♣r♊♣❡rt✐❡s✿ ✭✐✮ ❳ ❆❍ ❀❛s ✈❡r② ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❳ ❆❍ ✐s ❛ ❝♊♠♣❛❝t ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t②❀ ✶✷

  66. ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❳ ❆❍ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ ❆r❣②r♩s✕❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❚❀❡♊r❡♠ ✭❆r❣②r♩s✕❍❛②❞♊♥ ✷✵✶✶✮✳ ❚❀❡r❡ ✐s ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ✇❀✐❝❀ ❀❛s t❀❡ ❢♊❧❧♊✇✐♥❣ t❀r❡❡ ♣r♊♣❡rt✐❡s✿ ✭✐✮ ❳ ❆❍ ❀❛s ✈❡r② ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❳ ❆❍ ✐s ❛ ❝♊♠♣❛❝t ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳ ❆❍ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✶✷

  67. ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ ❆r❣②r♩s✕❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❚❀❡♊r❡♠ ✭❆r❣②r♩s✕❍❛②❞♊♥ ✷✵✶✶✮✳ ❚❀❡r❡ ✐s ❛ ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ✇❀✐❝❀ ❀❛s t❀❡ ❢♊❧❧♊✇✐♥❣ t❀r❡❡ ♣r♊♣❡rt✐❡s✿ ✭✐✮ ❳ ❆❍ ❀❛s ✈❡r② ❢❡✇ ♊♣❡r❛t♩rs✱ ✐♥ t❀❡ s❡♥s❡ t❀❛t ❡❛❝❀ ♊♣❡r❛t♩r ♊♥ ❳ ❆❍ ✐s ❛ ❝♊♠♣❛❝t ♣❡rt✉r❜❛t✐♊♥ ♊❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♊❢ t❀❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳ ❆❍ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❳ ❆❍ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ℓ ✶ ✳ ✶✷

  68. ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ✷ ✷ ✲♠❛tr✐❝❡s ❚ ✶ ✶ ❳ ❆❍ ❳ ❆❍ ❚ ✶ ✷ ❳ ❆❍ ❚ ✷ ✶ ❳ ❆❍ ❚ ✷ ✷ ❑❡② ♣♊✐♥t✿ ❚ ✶ ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✶ ✶ ✐s ❝♊♠♣❛❝t ❊ ✶ ❚ ✷ ✶ ❚ ✷ ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ ❊ ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ✶✞

  69. ❑❡② ♣♊✐♥t✿ ❚ ✶ ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✶ ✶ ✐s ❝♊♠♣❛❝t ❊ ✶ ❚ ✷ ✶ ❚ ✷ ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ ❊ ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ✶✞

  70. ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✶ ✶ ✐s ❝♊♠♣❛❝t ❊ ✶ ❚ ✷ ✶ ❚ ✷ ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ ❊ ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♊✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ✶✞

  71. ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♊✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♊♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ B ( ❊ ) ✶✞

  72. ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♊✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♊♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ✶✞

  73. ✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♊✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♊♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞ ✶✞

  74. ▌♊r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ✵ ✵ ▲ ❱❯ ❲ ✇❀❡r❡ ❳ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♊✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♊♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ✶✞

  75. ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■■✿ t❀❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♊♣❡r❛t♩rs ❚ ♊♥ ❊ ✇✐t❀ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s ï¿œ ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ ï¿œ ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♊✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ s❡t ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♊♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❝♊❞✐♠❡♥s✐♊♥ ♊♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❀❡♥❝❡ ❛❧s♩ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♊t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▌♊r❡ ♣r❡❝✐s❡❧②✱ K ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ ♊♣❡r❛t♩r ï¿œ ✵ ✵ ï¿œ ▲ = , ❱❯ ∗ κ ❲ ✇❀❡r❡ κ : ❳ ❆❍ → ❳ ∗∗ ❆❍ ✐s t❀❡ ❝❛♥♊♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❀✐❧❡ ❯ : ℓ ✶ → ❳ ∗ ❆❍ ✱ ❱ : ℓ ∗ ✶ = ℓ ∞ → ℓ ∞ ( ✷ N − ✶ ) ❛♥❞ ❲ : ℓ ∞ → ℓ ∞ ( ✷ N ) ❛r❡ ✐s♩♠♩r♣❀✐s♠s✳ ✶✞

  76. ❍❡♥❝❡ ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✷ ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r ❊ ❚ ✷ ✶ ❚ ✷ ✷ ✇❀✐❝❀ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ ❊ ✱ ✐s ♥♊t ❝♊♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ ❊ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❀❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ✐s ♣♊s✐t✐✈❡ ❢♊r ❊✳ ❍♩✇ ❛❜♩✉t ◗✉❡st✐♊♥ ■❄ ❘❡❝❛❧❧✿ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ ✐❞❡❛❧ K ✶ ✐s t❀❡ ✉♥✐q✉❡ ♥♊♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ✶✹

  77. ■♥ ♣❛rt✐❝✉❧❛r✱ t❀❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ✐s ♣♊s✐t✐✈❡ ❢♊r ❊✳ ❍♩✇ ❛❜♩✉t ◗✉❡st✐♊♥ ■❄ ❘❡❝❛❧❧✿ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ ✐❞❡❛❧ K ✶ ✐s t❀❡ ✉♥✐q✉❡ ♥♊♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✷ , ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r , ❚ ✷ , ✶ ❚ ✷ , ✷ ✇❀✐❝❀ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ✐s ♥♊t ❝♊♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ✶✹

  78. ❍♩✇ ❛❜♩✉t ◗✉❡st✐♊♥ ■❄ ❘❡❝❛❧❧✿ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❚❀❡♊r❡♠ ✭❉❑❑❑▲✮✳ ❚❀❡ ✐❞❡❛❧ K ✶ ✐s t❀❡ ✉♥✐q✉❡ ♥♊♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ❍❡♥❝❡ ￜￜ ❚ ✶ , ✶ ï¿œ ï¿œ ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✷ , ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r , ❚ ✷ , ✶ ❚ ✷ , ✷ ✇❀✐❝❀ ✐s ❛ ♠❛①✐♠❛❧ t✇♊✲s✐❞❡❞ ✐❞❡❛❧ ♊❢ B ( ❊ ) ✱ ✐s ♥♊t ❝♊♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❀❡ ❛♥s✇❡r t♩ ◗✉❡st✐♊♥ ■ ✐s ♣♊s✐t✐✈❡ ❢♊r ❊✳ ✶✹

  79. ✭❘❡❝❛❧❧✿ ✐♥ t❀❡ ❡①❛♠♣❧❡✱ ❛❜♊✈❡✱ ❊ ✳✮ ❳ ❆❍ ❆♥s✇❡r✿ ②❡s✩ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♩s ❛♥❞ ❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ❝♊♥t❛✐♥s ❛ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ❚ ♊❢ ✐♥✜♥✐t❡ ❝♊❞✐♠❡♥s✐♊♥ s✉❝❀ t❀❛t✿ ✭✐✮ ❡❛❝❀ ♊♣❡r❛t♩r ❢r♩♠ ❚ ✐♥t♩ ❳ ❆❍ ❀❛s t❀❡ ❢♊r♠ ❏ ❑ ❢♊r s♊♠❡ ❛♥❞ s♊♠❡ ❝♊♠♣❛❝t ♊♣❡r❛t♩r ❑✱ ✇❀❡r❡ ❏ ❚ ❳ ❆❍ ❞❡♥♊t❡s t❀❡ ✐♥❝❧✉s✐♊♥❀ ✭✐✐✮ ❚ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❚ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♊♥✿ ■s t❀❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ s✉❝❀ t❀❛t B ( ❊ ) ❝♊♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✶✺

  80. ❆♥s✇❡r✿ ②❡s✩ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♩s ❛♥❞ ❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ❝♊♥t❛✐♥s ❛ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ❚ ♊❢ ✐♥✜♥✐t❡ ❝♊❞✐♠❡♥s✐♊♥ s✉❝❀ t❀❛t✿ ✭✐✮ ❡❛❝❀ ♊♣❡r❛t♩r ❢r♩♠ ❚ ✐♥t♩ ❳ ❆❍ ❀❛s t❀❡ ❢♊r♠ ❏ ❑ ❢♊r s♊♠❡ ❛♥❞ s♊♠❡ ❝♊♠♣❛❝t ♊♣❡r❛t♩r ❑✱ ✇❀❡r❡ ❏ ❚ ❳ ❆❍ ❞❡♥♊t❡s t❀❡ ✐♥❝❧✉s✐♊♥❀ ✭✐✐✮ ❚ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❚ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♊♥✿ ■s t❀❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ s✉❝❀ t❀❛t B ( ❊ ) ❝♊♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❀❡ ❡①❛♠♣❧❡✱ ❛❜♊✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ✶✺

  81. ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♩s ❛♥❞ ❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ❝♊♥t❛✐♥s ❛ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ❚ ♊❢ ✐♥✜♥✐t❡ ❝♊❞✐♠❡♥s✐♊♥ s✉❝❀ t❀❛t✿ ✭✐✮ ❡❛❝❀ ♊♣❡r❛t♩r ❢r♩♠ ❚ ✐♥t♩ ❳ ❆❍ ❀❛s t❀❡ ❢♊r♠ ❏ ❑ ❢♊r s♊♠❡ ❛♥❞ s♊♠❡ ❝♊♠♣❛❝t ♊♣❡r❛t♩r ❑✱ ✇❀❡r❡ ❏ ❚ ❳ ❆❍ ❞❡♥♊t❡s t❀❡ ✐♥❝❧✉s✐♊♥❀ ✭✐✐✮ ❚ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❚ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♊♥✿ ■s t❀❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ s✉❝❀ t❀❛t B ( ❊ ) ❝♊♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❀❡ ❡①❛♠♣❧❡✱ ❛❜♊✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✩ ✶✺

  82. ✭✐✐✮ ❚ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❚ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♊♥✿ ■s t❀❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ s✉❝❀ t❀❛t B ( ❊ ) ❝♊♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❀❡ ❡①❛♠♣❧❡✱ ❛❜♊✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✩ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♩s ❛♥❞ ❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ❝♊♥t❛✐♥s ❛ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ❚ ♊❢ ✐♥✜♥✐t❡ ❝♊❞✐♠❡♥s✐♊♥ s✉❝❀ t❀❛t✿ ✭✐✮ ❡❛❝❀ ♊♣❡r❛t♩r ❢r♩♠ ❚ ✐♥t♩ ❳ ❆❍ ❀❛s t❀❡ ❢♊r♠ α ❏ + ❑ ❢♊r s♊♠❡ α ∈ C ❛♥❞ s♊♠❡ ❝♊♠♣❛❝t ♊♣❡r❛t♩r ❑✱ ✇❀❡r❡ ❏ : ❚ → ❳ ❆❍ ❞❡♥♊t❡s t❀❡ ✐♥❝❧✉s✐♊♥❀ ✶✺

  83. ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❚ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♊♥✿ ■s t❀❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ s✉❝❀ t❀❛t B ( ❊ ) ❝♊♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❀❡ ❡①❛♠♣❧❡✱ ❛❜♊✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✩ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♩s ❛♥❞ ❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ❝♊♥t❛✐♥s ❛ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ❚ ♊❢ ✐♥✜♥✐t❡ ❝♊❞✐♠❡♥s✐♊♥ s✉❝❀ t❀❛t✿ ✭✐✮ ❡❛❝❀ ♊♣❡r❛t♩r ❢r♩♠ ❚ ✐♥t♩ ❳ ❆❍ ❀❛s t❀❡ ❢♊r♠ α ❏ + ❑ ❢♊r s♊♠❡ α ∈ C ❛♥❞ s♊♠❡ ❝♊♠♣❛❝t ♊♣❡r❛t♩r ❑✱ ✇❀❡r❡ ❏ : ❚ → ❳ ❆❍ ❞❡♥♊t❡s t❀❡ ✐♥❝❧✉s✐♊♥❀ ✭✐✐✮ ❚ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✶✺

  84. ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♊♥✿ ■s t❀❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❀ s♣❛❝❡ ❊ s✉❝❀ t❀❛t B ( ❊ ) ❝♊♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❀❡ ❡①❛♠♣❧❡✱ ❛❜♊✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✩ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♩s ❛♥❞ ❍❛②❞♊♥✬s ❇❛♥❛❝❀ s♣❛❝❡ ❳ ❆❍ ❝♊♥t❛✐♥s ❛ ❝❧♩s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♊♥❛❧ s✉❜s♣❛❝❡ ❚ ♊❢ ✐♥✜♥✐t❡ ❝♊❞✐♠❡♥s✐♊♥ s✉❝❀ t❀❛t✿ ✭✐✮ ❡❛❝❀ ♊♣❡r❛t♩r ❢r♩♠ ❚ ✐♥t♩ ❳ ❆❍ ❀❛s t❀❡ ❢♊r♠ α ❏ + ❑ ❢♊r s♊♠❡ α ∈ C ❛♥❞ s♊♠❡ ❝♊♠♣❛❝t ♊♣❡r❛t♩r ❑✱ ✇❀❡r❡ ❏ : ❚ → ❳ ❆❍ ❞❡♥♊t❡s t❀❡ ✐♥❝❧✉s✐♊♥❀ ✭✐✐✮ ❚ ❀❛s ❛ ❙❝❀❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❀❡ ❞✉❛❧ s♣❛❝❡ ♊❢ ❚ ✐s ✐s♩♠♩r♣❀✐❝ t♩ ℓ ✶ ✳ ✶✺

  85. ❜✉t ✶ ✐s ♥♊t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♊♣❡r❛t♩r ♊♥ ❊ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❀❡r❡ ❛r❡ ❡①❛❝t❧② t✇♩ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ ❊ ✱ ♥❛♠❡❧② ✵ ✵ ❚ ❊ ❛♥❞ ❚ ❊ ✶ ✷ ✷ ✷ ✶ ✶ ✭✐✐✮ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ t✇♩ ♊♣❡r❛t♩rs ✵ ✵ ■ ❳ ❆❍ ❏ ❛♥❞ ✵ ✵ ✵ ✵ ✭✐✐✐✮ ✷ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♊♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❚ ✳ ❚❀❡♥ ❡❛❝❀ ❚ ∈ B ( ❊ ) ❀❛s t❀❡ ❢♊r♠ ï¿œ α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ ï¿œ α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❚ + ❑ ✷ , ✷ ✇❀❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❀❡ ♊♣❡r❛t♩rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♊♠♣❛❝t✳ ✶✻

  86. ❜✉t ✶ ✐s ♥♊t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♊♣❡r❛t♩r ♊♥ ❊ ✭✐✐✮ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ t✇♩ ♊♣❡r❛t♩rs ✵ ✵ ■ ❳ ❆❍ ❏ ❛♥❞ ✵ ✵ ✵ ✵ ✭✐✐✐✮ ✷ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♊♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❚ ✳ ❚❀❡♥ ❡❛❝❀ ❚ ∈ B ( ❊ ) ❀❛s t❀❡ ❢♊r♠ ï¿œ α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ ï¿œ α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❚ + ❑ ✷ , ✷ ✇❀❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❀❡ ♊♣❡r❛t♩rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❀❡r❡ ❛r❡ ❡①❛❝t❧② t✇♩ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✶✻

  87. ❜✉t ✶ ✐s ♥♊t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♊♣❡r❛t♩r ♊♥ ❊ ✭✐✐✐✮ ✷ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♊♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❚ ✳ ❚❀❡♥ ❡❛❝❀ ❚ ∈ B ( ❊ ) ❀❛s t❀❡ ❢♊r♠ ï¿œ α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ ï¿œ α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❚ + ❑ ✷ , ✷ ✇❀❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❀❡ ♊♣❡r❛t♩rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❀❡r❡ ❛r❡ ❡①❛❝t❧② t✇♩ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✭✐✐✮ M ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ t✇♩ ♊♣❡r❛t♩rs ✵ ï¿œ ✵ ï¿œ ■ ❳ ❆❍ ï¿œ ï¿œ ❏ ❛♥❞ , ✵ ✵ ✵ ✵ ✶✻

  88. ✭✐✐✐✮ ✷ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♊♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❚ ✳ ❚❀❡♥ ❡❛❝❀ ❚ ∈ B ( ❊ ) ❀❛s t❀❡ ❢♊r♠ ï¿œ α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ ï¿œ α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❚ + ❑ ✷ , ✷ ✇❀❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❀❡ ♊♣❡r❛t♩rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❀❡r❡ ❛r❡ ❡①❛❝t❧② t✇♩ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✭✐✐✮ M ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ t✇♩ ♊♣❡r❛t♩rs ✵ ï¿œ ✵ ï¿œ ■ ❳ ❆❍ ï¿œ ï¿œ ❏ ❛♥❞ , ✵ ✵ ✵ ✵ ❜✉t M ✶ ✐s ♥♊t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♊♣❡r❛t♩r ♊♥ ❊ ; ✶✻

  89. ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♊♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❚ ✳ ❚❀❡♥ ❡❛❝❀ ❚ ∈ B ( ❊ ) ❀❛s t❀❡ ❢♊r♠ ï¿œ α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ ï¿œ α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❚ + ❑ ✷ , ✷ ✇❀❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❀❡ ♊♣❡r❛t♩rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♊♠♣❛❝t✳ ❚❀❡♊r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❀❡r❡ ❛r❡ ❡①❛❝t❧② t✇♩ ♥♊♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✭✐✐✮ M ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❀❡ t✇♩ ♊♣❡r❛t♩rs ✵ ï¿œ ✵ ï¿œ ■ ❳ ❆❍ ï¿œ ï¿œ ❏ ❛♥❞ , ✵ ✵ ✵ ✵ ❜✉t M ✶ ✐s ♥♊t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♊♣❡r❛t♩r ♊♥ ❊ ; ✭✐✐✐✮ M ✷ ✐s ♥♊t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ✶✻

  90. ❛♥❞ r❡❢❡r❡♥❝❡s ❲❀❛t ✐s t❀❡ s✐t✉❛t✐♊♥ ❢♊r ♠❛①✐♠❛❧ r✐❣❀t ✐❞❡❛❧s ♊❢ ❊ ❄ ❑❡② r❡❢❡r❡♥❝❡s ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♊❝❀❛♥❡❊✱ P✳ ❑♩s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▌❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ t❀❡ ❇❛♥❛❝❀ ❛❧❣❡❜r❛ ♊❢ ❜♊✉♥❞❡❞ ♊♣❡r❛t♩rs ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡✱ ❙t✉❞✐❛ ▌❛t❀✳ ✷✶✜ ✭✷✵✶✞✮✱ ✷✹✺✕✷✜✻✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❊♊✱ ●❡♥❡r❛t♩rs ♊❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❀ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▌❛t❀✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✌✞✕✶✟✞✳ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♊❢ t❀❡ ❛❧❣❡❜r❛ ♊❢ ❜♊✉♥❞❡❞ ♊♣❡r❛t♩rs ❛❝t✐♥❣ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♊♥✳ ❖♣❡♥ ♣r♊❜❧❡♠s ◮ ▲❡t ❊ = ❈ ( ❑ ) ✱ ✇❀❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❀ t❀❛t ❈ ( ❑ ) ￜ∌ = ❝ ✵ ✳ ■s ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ✶✌

  91. ❛♥❞ r❡❢❡r❡♥❝❡s ❑❡② r❡❢❡r❡♥❝❡s ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♊❝❀❛♥❡❊✱ P✳ ❑♩s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▌❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♊❢ t❀❡ ❇❛♥❛❝❀ ❛❧❣❡❜r❛ ♊❢ ❜♊✉♥❞❡❞ ♊♣❡r❛t♩rs ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡✱ ❙t✉❞✐❛ ▌❛t❀✳ ✷✶✜ ✭✷✵✶✞✮✱ ✷✹✺✕✷✜✻✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❊♊✱ ●❡♥❡r❛t♩rs ♊❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❀ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▌❛t❀✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✌✞✕✶✟✞✳ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♊❢ t❀❡ ❛❧❣❡❜r❛ ♊❢ ❜♊✉♥❞❡❞ ♊♣❡r❛t♩rs ❛❝t✐♥❣ ♊♥ ❛ ❇❛♥❛❝❀ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♊♥✳ ❖♣❡♥ ♣r♊❜❧❡♠s ◮ ▲❡t ❊ = ❈ ( ❑ ) ✱ ✇❀❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♊♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❀ t❀❛t ❈ ( ❑ ) ￜ∌ = ❝ ✵ ✳ ■s ❡❛❝❀ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♊❢ B ( ❊ ) ✜①❡❞❄ ◮ ❲❀❛t ✐s t❀❡ s✐t✉❛t✐♊♥ ❢♊r ♠❛①✐♠❛❧ r✐❣❀t ✐❞❡❛❧s ♊❢ B ( ❊ ) ❄ ✶✌

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