# Equações Diferenciais Hereditárias Estocásticas e o Problema de Portfólio Ótimo

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## Texto

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### ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■▼

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

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### ▼❛r✐❛♥❛ ❙✐❧✈❛ ❚❛✈❛r❡s

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❊❞s♦♥ ❆❧❜❡rt♦ ❈♦❛②❧❛ ❚❡r❛♥✳

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❚❛✈❛r❡s✱ ▼❛r✐❛♥❛ ❙✐❧✈❛✳

❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❍❡r❡❞✐tár✐❛s ❊st♦❝ást✐❝❛s ❡ ♦ Pr♦❜❧❡♠❛ ❞❡ P♦rt❢ó❧✐♦ Ót✐♠♦ ✴ ▼❛r✐❛♥❛ ❙✐❧✈❛ ❚❛✈❛r❡s✳ ✕ ✷✵✶✹✳

✽✷ ❢✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❊❞s♦♥ ❆❧❜❡rt♦ ❈♦❛②❧❛ ❚❡r❛♥✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❙❛❧✈❛❞♦r✱ ✷✵✶✹✳

✶✳ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡st♦❝ást✐❝❛s✳ ✷✳ ▼❛t❡♠át✐❝❛ ✜♥❛♥✲ ❝❡✐r❛✳ ■♥✈❡st✐♠❡♥t♦s ✲ ▼❛t❡♠át✐❝❛✳ ■✳ ❚❡r❛♥✱ ❊❞s♦♥ ❆❧❜❡rt♦ ❈♦❛②❧❛✳ ■■✳ ❯♥✐✈❡rs✐s❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ■■■✳ ❚ít✉❧♦✳

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### ▼❛r✐❛♥❛ ❙✐❧✈❛ ❚❛✈❛r❡s

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✶✹ ❞❡ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✹✳

### ❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❊❞s♦♥ ❆❧❜❡t♦ ❈♦❛②❧❛ ❚❡r❛♥ ✭❖r✐❡♥t❛❞♦r✮ ❯❋❇❆

Pr♦❢✳ ❉r✳ ▼❛♥✉❡❧ ❙t❛❞❧❜❛✉❡r ❯❋❇❆

Pr♦❢❛✳ ❉r✳ ●✐♦✈❛♥❛ ❖❧✐✈❡✐r❛ ❙✐❧✈❛

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## ❆❣r❛❞❡❝✐♠❡♥t♦s

❆♥t❡s ❞❡ q✉❛❧q✉❡r ♠❡♥s❛❣❡♠✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛✱ ♣❛❝✐ê♥❝✐❛ ❡ ❢é ♣❛r❛ r❡❛❧✐③❛r ❡ss❡ tr❛❜❛❧❤♦✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠❛♠✐✱ ▼❛r✐❛✱ ♣♦r t♦❞♦ ❛♠♦r ❝♦♥❝❡❞✐❞♦✳ ❈❤❡❣❛r ❡ s❛✐r ❞❡ ❝❛s❛ ❝♦♠ ❛ ❝❡rt❡③❛ q✉❡ t❡♠ ❛❧❣✉é♠ ❝✉✐❞❛♥❞♦ ❞❡ t✉❞♦ ♣❛r❛ t❡ ❞❡✐①❛r ❜❡♠✱ ♥ã♦ t❡♠ ♣r❡ç♦✳ ❖❜r✐❣❛❞❛ ♠ã❡✦ ❆♦ ♠❡✉ ♣❛♣✐✱ ❊❞✱ q✉❡ ♠❡ ❛❥✉❞♦✉ ❛ t♦♠❛r ❞❡❝✐sõ❡s ❝♦♠ s❡❣✉r❛♥ç❛ ❡✱ ♠❡s♠♦ s❡♠ s❛❜❡r✱ ♠❡ ❞❡✉ ❝❛r✐♥❤♦ q✉❛♥❞♦ ❡✉ ♠❛✐s ♣r❡❝✐s❛✈❛✳ ❆♦ ♠❡✉ ✐r♠ã♦ ❊❞✉❛r❞♦✱ ♠❡ ❡♥❝❤❡♥❞♦ ❛ ♣❛❝✐ê♥❝✐❛ ❛té ❛rr❛♥❝❛r ✉♠ s♦rr✐s♦✳ ▼✐♥❤❛ ✈ó③✐♥❤❛✱ ■s❛❜❡❧✱ s✐♠♣❧❡s♠❡♥t❡ ♣♦r ❡①✐st✐r ❡ s❡r tã♦ ✈♦❝ê✦ ❚♦❞♦ ❡ss❡ ❛❣r❛❞❡❝✐♠❡♥t♦ é ❡①t❡♥s✐✈♦ ❛♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✳ ❱♦❝ês ✈❛❧❡♠ ♦✉r♦✳

❊①♣r✐♠♦ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ✐♠❡♥s❛ ❛ ❆❞❡r❜❛❧✱ ♠❡✉ ❝♦♠♣❛♥❤❡✐r♦ ❡ ❛♠✐❣♦✳ ❙❡♠ ✈♦❝ê✱ ♥ã♦ ❝❤❡❣❛r✐❛ ❛té ❛q✉✐✳ ❱♦❝ê ❢♦✐ ♠❡✉ ❛❧✐❝❡r❝❡ ❡♠ ❝❛❞❛ ♠♦♠❡♥t♦ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ✷ ❛♥♦s✳ ❖❜r✐❣❛❞❛ ♣♦r t✉❞♦✱ ♠❡✉ ❛♠♦r✳ ❚❊ ❆▼❖✳ ❚✐❛ ❘❡❣❡✱ ♠✐♥❤❛ s♦❣r✐t❛✱ ♠✐♥❤❛ s❡❣✉♥❞❛ ♠ã❡③✐♥❤❛✱ ❞❡✐①♦ ✉♠ ♠✉✐t♦ ♦❜r✐❣❛❞❛ ♣♦r ❝✉✐❞❛r ❞❡ ♠✐♠ ❝♦♠ t❛♥t♦ ❛♣r❡ç♦✳

❊t❡r♥❛♠❡♥t❡ ❛❣r❛❞❡❝✐❞❛ ❛ ■s✐s✱ q✉❡r✐❞❛ ♣r✐♠❛ q✉❡ ❛ t♦❞♦s ♠♦♠❡♥t♦s ❡stá ❞✐s♣♦st❛ ❛ ♠❡ ♦✉✈✐r ❡ ♠❡ ❞❛r ❛♠♦r✳ ❚❛✐s❡✱ q✉❡ ❝♦♠♦ ❞❡♠♦♥str❛çã♦ ❞❛ s✉❛ ✜❡❧ ❛♠✐③❛❞❡✱ ♠❡ ❞❡✉ ♦ ♠❛✐♦r ♣r❡s❡♥t❡ q✉❡ ♣♦❞✐❛ t❡r r❡❝❡❜✐❞♦✳ ❙❡r ♠❛❞r✐♥❤❛ ❞❡ ❙♦♣❤✐❛ ♠❡ ❢❡③ s❡♥t✐r ♦ ❛♠♦r ♠❛✐s ♣✉r♦ ❡ ✈ê✲❧❛ s♦rr✐r s❡♠♣r❡ ❡♥❝❤❡ ♠❡✉ ❝♦r❛çã♦ ❞❡ ❡s♣❡r❛♥ç❛✳ ❩✐♥❤❛ ❡ ❚❛✐✱ ✈♦❝ês sã♦ ❝♦♠♦ ✐r♠ãs ♣❛r❛ ♠✐♠✳ ❆♦ ♠❡✉ ✐r♠ã♦③✐♥❤♦ ❞❡ ❝♦r❛çã♦✱ ■♥❤♦✱ q✉❡ ♠❡s♠♦ ♥❛ ❆❧❡♠❛♥❤❛ ♥ã♦ ♠❡ ❡sq✉❡❝❡✳ ❚✐❛ ❚❡r❡③❛✱ t✐❛ ❙❡❧♠❛ ❡ t✐❛ ❉❡♥❡✱ ♦❜r✐❣❛❞❛ ♣♦r ✈♦❝ês ♠❡ ❛❝❛❧❡♥t❛r❡♠ s❡♠♣r❡✳ ❊✉ ❛❞♦r♦ ❞❡♠❛✐s ✈♦❝ês✳

❆❣r❛❞❡❝✐♠❡♥t♦ ♠❛✐s ❞♦ q✉❡ ♠❡r❡❝✐❞♦ ♣❛r❛ ♠❡✉ ♣r♦❢❡ss♦r✱ ❊❞s♦♥✱ ♣♦r t♦❞♦ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ❝♦♠♣❛rt✐❧❤❛❞♦ ❡ t♦❞❛ ❞❡❞✐❝❛çã♦ ♣❛r❛ ♠❡ ❛❥✉❞❛r ❛ s✉♣❡r❛r ♦s ♦❜stá❝✉❧♦s ❞❛ ❞✐ss❡rt❛çã♦✳ ❖ ♠❡✉ ❝❛r✐♥❤♦ t❛♠❜é♠ ❛♦ ♣r♦❢❡ss♦r ❘❛②♠✉♥❞♦ ❚♦rr❡s✱ ♣♦r s❡r tã♦ s♦❧í❝✐t♦✱ ❡ ❛ ♣r♦❢❡ss♦r❛ ❘✐t❛ ❞❡ ❈áss✐❛✱ ♣♦r s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r ❡ ❛❝r❡❞✐t❛r ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧✳

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✏➱ ♠❡❧❤♦r ❧❛♥ç❛r✲s❡ à ❧✉t❛ ❡♠ ❜✉s❝❛ ❞♦ tr✐✉♥❢♦✱ ♠❡s♠♦ ❡①♣♦♥❞♦✲s❡ ❛♦ ✐♥s✉❝❡ss♦✱ ❞♦ q✉❡ ✜❝❛r ♥❛ ✜❧❛ ❞♦s ♣♦❜r❡s ❞❡ ❡s♣ír✐t♦✱ q✉❡ ♥❡♠ ❣♦③❛♠ ♠✉✐t♦ ♥❡♠ s♦❢r❡♠ ♠✉✐t♦✱ ♣♦r ✈✐✲ ✈❡r❡♠ ♥❡ss❛ ♣❡♥✉♠❜r❛ ❝✐♥③❡♥t❛ ❞❡ ♥ã♦ ❝♦✲ ♥❤❡❝❡r ✈✐tór✐❛ ❡ ♥❡♠ ❞❡rr♦t❛✑

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## ❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ❞❡ ♣♦rt❢ó✲ ❧✐♦✳ ❖ ♣♦rt❢ó❧✐♦ s❡rá ❝♦♠♣♦st♦ ♣♦r ❞♦✐s t✐♣♦s ❞❡ ✐♥✈❡st✐♠❡♥t♦✿ ✉♠ s❡♠ r✐s❝♦✱ q✉❡ s❡rá ✉♠❛ ❝♦♥t❛ ♣♦✉♣❛♥ç❛✱ ❡ ♦✉tr♦ ❝♦♠ r✐s❝♦✱ q✉❡ s❡rá ✉♠❛ ❝♦♥t❛ ❞❡ ❛çõ❡s ♥♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦✳ ❖ ♣r❡ç♦ ❞❛s ❛çõ❡s s❡rá ♠♦❞❡❧❛❞♦ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡st♦❝ást✐❝❛ ❤❡r❡❞✐tár✐❛✱ ❡ ♦ ♦❜❥❡t✐✈♦ ❞♦ tr❛❜❛❧❤♦ s❡rá ❡♥❝♦♥tr❛r ✉♠❛ ❡str❛té❣✐❛ ❞❡ ❝♦♥s✉♠♦✲♥❡❣♦❝✐❛çã♦ q✉❡ ♠❛①✐♠✐③❡ ♦ ❢✉♥❝✐♦♥❛❧ ❝♦♥s✉♠♦✱ s❡♠ ❝❛✉s❛r ❞é✜❝✐t ♥❛ ❝♦♥t❛ ♣♦✉♣❛♥ç❛✳

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## ❆❜str❛❝t

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ✐♥t❡♥❞s t♦ st✉❞② t❤❡ ♣r♦❜❧❡♠ ♦❢ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥✳ ❚❤❡ ♣r♦❜❧❡♠ ❝♦♥s✐sts ♦❢ t✇♦ ✐♥✈❡st♠❡♥t t②♣❡s✿ ❛ s❛❢❡ ♦♥❡✱ ✇❤✐❝❤ ✐s ❛ s❛✈✐♥❣s ❛❝❝♦✉♥t✱ ❛♥❞ ❛ r✐s❦② ♦♥❡✱ ✇❤✐❝❤ ✐s ❛♥ ❛❝❝♦✉♥t ♦❢ s❤❛r❡s ✐♥ t❤❡ ✜♥❛♥❝✐❛❧ ♠❛r❦❡t✳ ❚❤❡ s❤❛r❡s ♣r✐❝❡ ✐s ♠♦❞❡❧❡❞ ❜② ❛ ❤❡r❡❞✐t❛r② st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ❛♥❞ t❤❡ ♠❛✐♥ ❣♦❛❧ ✐s t♦ ✜♥❞ ❛ tr❛♥❞✐♥❣✲❝♦♥s✉♠♣t✐♦♥ str❛t❡❣② t❤❛t ♠❛①✐♠✐③❡s t❤❡ ❝♦♥s✉♠♣t✐♦♥ ❢✉♥❝t✐♦♥❛❧ ❧❡❛✈✐♥❣ ♥♦ ❞❡✜❝✐t ✐♥ t❤❡ s❛✈✐♥❣s ❛❝❝♦✉♥t✳

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## ❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✸

✶✳✶ Pr❡❧✐♠✐♥❛r❡s Pr♦❜❛❜✐❧íst✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❊s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷ ▼♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ❡ ■♥t❡❣r❛❧ ❞❡ ■tô ✷✷ ✷✳✶ ▼❛rt✐♥❣❛❧❡ ❡ ❚❡♠♣♦ ❞❡ P❛r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ▼♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ■♥t❡❣r❛❧ ❊st♦❝ást✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❊st♦❝ást✐❝❛ ❍❡r❡❞✐tár✐❛ ❝♦♠ ▼❡♠ór✐❛ ■❧✐♠✐t❛❞❛ ✹✵ ✸✳✶ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹ ❖t✐♠✐③❛çã♦ ❞❡ P♦rt❢ó❧✐♦ ❍❡r❡❞✐tár✐♦ ✺✷

✹✳✶ ❖ Pr♦❜❧❡♠❛ ❞❡ ❖t✐♠✐③❛çã♦ ❞❡ P♦rt❢ó❧✐♦ ❍❡r❡❞✐tár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✶✳✶ ❊str✉t✉r❛ ❞❡ Pr❡ç♦ ❍❡r❡❞✐tár✐♦ ❝♦♠ ▼❡♠ór✐❛ ■❧✐♠✐t❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✶✳✷ ❖ ❡s♣❛ç♦ ❞♦s ✐♥✈❡♥tár✐♦s ❞❛s ❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✶✳✸ ❊str❛té❣✐❛s ❞❡ ❝♦♥s✉♠♦✲♥❡❣♦❝✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✶✳✹ ❘❡❣✐ã♦ ❞❡ ❙♦❧✈ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✶✳✺ ❉✐♥â♠✐❝❛ ❞♦ P♦rt❢ó❧✐♦ ❡ ❊str❛té❣✐❛s ❆❞♠✐ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✶✳✻ ❋♦r♠✉❧❛çã♦ ❞♦ Pr♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✷ ❖ Pr♦❝❡ss♦ ❈♦♥tr♦❧❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✸ ❆ ❍❏❇◗❱■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✸✳✶ ❖ Pr✐♥❝í♣✐♦ ❞❛ Pr♦❣r❛♠❛çã♦ ❉✐♥â♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✸✳✷ ❉❡❞✉çã♦ ❞❛ ❍❏❇◗❱■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸✳✸ ❱❛❧♦r❡s ❞❡ ❢r♦♥t❡✐r❛ ❞❛ ❍❏❇◗❱■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❱❡r✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

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## ■♥tr♦❞✉çã♦

❊st❡ tr❛❜❛❧❤♦ tr❛t❛rá ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ❞❡ ♣♦rt❢ó❧✐♦ ❤❡r❡❞✐tár✐♦ ❝♦♠ t❡♠♣♦ ✐♥✜♥✐t♦ ♥♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❝♦♥t❛ ♣♦✉♣❛♥ç❛ ❡ ✉♠❛ ❝♦♥t❛ ❞❡ ❛çõ❡s✳ ❱❛♠♦s s✉♣♦r q✉❡ ❛ ❝♦♥t❛ ♣♦✉♣❛♥ç❛ t❡♠ ❥✉r♦ ❝♦♠♣♦st♦ ❝♦♥tí♥✉♦ ❡ ♦ ♣r♦❝❡ss♦ ♣r❡ç♦ ✉♥✐tár✐♦ s❡❣✉❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♥ã♦ ❧✐♥❡❛r ❡st♦❝ást✐❝❛ ❤❡r❡❞✐tár✐❛ ❝♦♠ ✉♠❛ ♠❡♠ór✐❛ ✐♥✜♥✐t❛✱ ♠❛s ❞❡s❛♣❛r❡❝❡♥❞♦✳ ◆❛ ❞✐♥â♠✐❝❛ ❞❡ ♣r❡ç♦s ❞❡ ❛çõ❡s✱ ❛ss✉♠✐r❡♠♦s ❞✉❛s ❢✉♥çõ❡s✿ ❛ q✉❡ r❡♣r❡s❡♥t❛ ❛ t❛①❛ ❞❡ r❡t♦r♥♦ ♠é❞✐♦ ❡ ❛ q✉❡ r❡♣r❡s❡♥t❛ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❞♦s ♣r❡ç♦s ❞❛s ❛çõ❡s✱ q✉❡ ✈ã♦ ❞❡♣❡♥❞❡r ❞❡ t♦❞❛ ❛ ❤✐stór✐❛ ❞♦s ♣r❡ç♦s ❞❛s ❛çõ❡s s♦❜r❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦(−∞, t] ❡♠ ✈❡③ ❞❡ ❞❡♣❡♥❞❡r ❛♣❡♥❛s ❞♦ ♣r❡ç♦ ❛t✉❛❧ ♥♦ t❡♠♣♦ t0✳

❚r❛❜❛❧❤❛r❡♠♦s ♥✉♠❛ r❡❣✐ã♦ ❝❤❛♠❛❞❛ r❡❣✐ã♦ ❞❡ s♦❧✈ê♥❝✐❛✱ s♦❜ ♦s r❡q✉✐s✐t♦s ❞❡ ♣❛❣❛♠❡♥t♦ ✜①♦ ♠❛✐s ♣r♦♣♦r❝✐♦♥❛❧ ❞♦s ❝✉st♦s ❞❡ tr❛♥s❛çã♦ ❡ ✐♠♣♦st♦s ❞❡ ❣❛♥❤♦ ❞❡ ❝❛♣✐t❛❧✳ ❖ ✐♥✈❡st✐❞♦r s❡rá ♣❡r♠✐t✐❞♦ ❝♦♥s✉♠✐r ❞❛ s✉❛ ❝♦♥t❛ ♣♦✉♣❛♥ç❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠ ♣r♦❝❡ss♦ t❛①❛ ❞❡ ❝♦♥s✉♠♦ ❡ ♣♦❞❡rá ❢❛③❡r tr❛♥s❛çõ❡s ❡♥tr❡ s✉❛s ❝♦♥t❛s ♣♦✉♣❛♥ç❛ ❡ ❞❡ ❛çõ❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠❛ ❡str❛té❣✐❛ ❞❡ ♥❡❣♦❝✐❛çã♦✳ ❖ ✐♥✈❡st✐❞♦r ✈❛✐ s❡❣✉✐r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ r❡❣r❛s ❞❡ ❝♦♥s✉♠♦✱ tr❛♥s❛çã♦ ❡ tr✐❜✉t❛çã♦✳

◆❡st❡ tr❛❜❛❧❤♦✱ q✉❡r❡♠♦s ❜✉s❝❛r ✉♠❛ ❡str❛té❣✐❛ ❞❡ ❝♦♥s✉♠♦✲♥❡❣♦❝✐❛çã♦ ✐❞❡❛❧ ♣❛r❛ ♦ ✐♥✈❡st✐❞♦r✱ ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ♠❛①✐♠✐③❛r ❛ ✉t✐❧✐❞❛❞❡ ❡s♣❡r❛❞❛ ❛ ♣❛rt✐r ❞♦ ❝♦♥✲ s✉♠♦ t♦t❛❧ ❞❡ ❞❡s❝♦♥t♦✳ ❊st❡ ❞♦❝✉♠❡♥t♦ ❝♦♥té♠ ♦ t❡♦r❡♠❛ ❞❡ ✈❡r✐✜❝❛çã♦ ♣❛r❛ ❛ ♠❡❧❤♦r ❡str❛té❣✐❛✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ ✐♥tr♦❞✉③✐r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❢✉t✉r♦✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❞❡✜♥✐r ❛ ✐♥t❡❣r❛❧ ❡st♦❝ást✐❝❛✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❡st✉❞❛r❡♠♦s ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❤❡r❡❞✐tár✐❛ ❡st♦❝ást✐❝❛ ❝♦♠ ♠❡♠ór✐❛ ✐❧✐♠✐t❛❞❛ ♠❛s ❞❡s❛♣❛r❡❝❡♥❞♦✳ ❖ r❡s✉❧t❛❞♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♦❜t✐❞♦ ♥❡st❡ ❝❛♣ít✉❧♦ é ♦ t❡♦r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❡st❡ t✐♣♦ ❞❡ ❡q✉❛çã♦ é ♠❛✐s r❡❛❧íst✐❝♦✱ ♣♦r ❡♥✈♦❧✈❡r ♦ ♣❛ss❛❞♦✳ ❊①♣❧✐❝❛✲s❡ ❡♥tã♦ ♦ ♣♦rq✉ê ❞❡ ♠♦❞❡❧❛r♠♦s ♦s ♣r❡ç♦s ❞❛s ❛çõ❡s ❛tr❛✈és ❞❡st❛ ❡q✉❛çã♦✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ❡①✐❜✐r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ♣♦rt❢ó❧✐♦ ót✐♠♦✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡❞✉③✐✲ r❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ q✉❛s✐ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❍❛♠✐❧t♦♥ ❏❛❝♦❜✐ ❇❡❧❧♠❛♥✱ ❡st❛ q✉❡ ♥♦s ❞á ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s q✉❡ ❛ ❢✉♥çã♦ ✈❛❧♦r ❞❡✈❡ s❛t✐s❢❛③❡r✳ P♦r ✜♠✱ ❝❤❡❣❛r❡♠♦s ❛♦ ❚❡♦r❡♠❛ ❞❡ ❱❡r✐✜❝❛çã♦ q✉❡ ♥♦s ❢♦r♥❡❝❡ ❛s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s q✉❡ ❛ ❢✉♥çã♦ ✈❛❧♦r ❞❡✈❡ ❛t❡♥❞❡r✳

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## Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❡❧❡♠❡♥t❛r❡s q✉❡ s❡rã♦ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

### ✶✳✶ Pr❡❧✐♠✐♥❛r❡s Pr♦❜❛❜✐❧íst✐❝❛s

❙❡❥❛ Ω✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ t❛❧✈❡③ ❝♦♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ❞❡ três ♠♦❡❞❛s s❡♥❞♦ ❧❛♥ç❛❞❛s✳ Ω = {KKK, KKT, KT K, KT T, T KK, T KT, T T K, T T T}✱ ❑ r❡♣r❡s❡♥t❛ ❝♦r♦❛ ❡ ❚ r❡✲

♣r❡s❡♥t❛ ❝❛r❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❙❡❥❛ Ω 6= ✳ ❯♠❛ σ✲á❧❣❡❜r❛✱ F✱ é ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮ ∅ ∈ F

✭✐✐✮ ❙❡A ∈ F✱ ❡♥tã♦ AC

∈ F✳

✭✐✐✐✮ ❙❡ {Ai}i ⊂ F✱ ❡♥tã♦ S i

Ai ∈ F✳

❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❖❧❤❛♥❞♦ ♣❛r❛ Ω ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ✶✳✶✳✶✱ t❡♠♦s q✉❡

F0 ={∅,Ω}

F1 ={∅,Ω,{KKK, KKT, KT K, KT T},{T T T, T T K, T KT, T KK}}

F2 = t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω

sã♦ σ✲á❧❣❡❜r❛s✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❙❡❥❛ F ✉♠❛ σ✲á❧❣❡❜r❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✳ ❈❤❛♠❛♠♦s ❞❡ ♣r♦❜❛❜✐❧✐✲

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✭✐✮ P(Ω) = 1

✭✐✐✮ ❙❡{Ai}i◆ ⊂ F✱ ❡♥tã♦ P

S

i

Ai

≤ P

i

P(Ai)✳

✭✐✐✐✮ ❙❡ {Ai}i∈◆ ⊂ F✱ ❝♦♠ Ai∩Aj =∅✱ i6=j✱ ❡♥tã♦ P

S

i

Ai

= P

i

P(Ai)✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✺✳ ❙❡ A, B ∈ F✱ ❝♦♠ A B✱ ❡♥tã♦ P(A) P(B)✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ B =B\A∩A✱ ❡♥tã♦ P(B) =P(B\A) +P(A)✳ P♦rt❛♥t♦✱ P(B)≥P(A)✳

❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❯♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ é ✉♠❛ tr✐♣❧❛ (Ω,F,P) ♦♥❞❡ Ω é ✉♠ ❝♦♥✲ ❥✉♥t♦ ❞❛❞♦✱F ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡Ω✱ q✉❡ é ✉♠❛ σ✲á❧❣❡❜r❛✱ P ✉♠❛ ♠❡❞✐❞❛ ❞❡

♣r♦❜❛❜✐❧✐❞❛❞❡ ✭s♦❜r❡ F✮✳

❚❡r♠✐♥♦❧♦❣✐❛✿

• Ω é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❛♠♦str❛❧

• ♦s ❡❧❡♠❡♥t♦s A ❞❡F sã♦ ❝❤❛♠❛❞♦s ❡✈❡♥t♦s • ♦s ♣♦♥t♦s wΩsã♦ ❝❤❛♠❛❞♦s ♣♦♥t♦s ❛♠♦str❛✐s

• P(A)é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ❡✈❡♥t♦ A✳

• ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ é ✈❡r❞❛❞❡ ❡①❝❡t♦ ♣❛r❛ ✉♠ ❡✈❡♥t♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ③❡r♦ é

❝❤❛♠❛❞❛ ❛ s❡r s❛t✐s❢❡✐t❛ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳

❊①❡♠♣❧♦ ✶✳✶✳✼✳ ❙❡❥❛ Ω = {w1, w2, . . . , wN} ✜♥✐t♦ ❡ s✉♣♦♥❤❛ q✉❡ ❞❛♠♦s ♥ú♠❡r♦s 0 ≤

pj ≤ 1 ♣❛r❛ 1 ≤ j ≤ N✱ t❛❧ q✉❡ N

P

j=1

pj = 1✳ ❚♦♠❛♠♦s ❝♦♠♦ F t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡

Ω✳ P❛r❛ ❝❛❞❛ A ={wj1, wj2, . . . , wjn} ∈ F✱ ❝♦♠ 1≤j1 ≤ j2 ≤ . . .≤jn≤ N✳ ❉❡✜♥✐♠♦s P(A) :=pj1 +. . .+pjn✳

❖❜s❡r✈❡♠♦s q✉❡ (Ω,F,P) é ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ P(Ω) = 1✳ ❙❡

{Ai}M≤2

N

i=1 , Ak∩Aj =∅, k 6=j.

A1 ={wj11, wj12, . . . , wj1n} A2 ={wj21, wj22, . . . , wj2n}

· · ·

Ai ={wji1, wji2, . . . , wjin} ▲♦❣♦

M

[

i=1

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❉❛í✱

P

M

[

i=1

Ai

!

=pj1 +pj2 +. . .+pj(1n+2n+...+Mn) =P(A1) +. . .+P(AM) =

M

X

i=1

P(Ai).

❆❧é♠ ❞✐st♦✱ s❡ t♦♠❛r♠♦s Bi✬s ❞✐s❥✉♥t♦s t❛✐s q✉❡ Bi ⊂Ai✱ ∀i✱ ❡ M

S

i=1

Ai = M

S

i=1

Bi✱

P

M

[

i=1

Ai

!

=P

M

[

i=1

Bi

!

=P(B1) +. . .+P(BM)≤P(A1) +. . .P(An).

❉❡✜♥✐çã♦ ✶✳✶✳✽✳ ❆σ✲á❧❣❡❜r❛ q✉❡ ❝♦♥té♠ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❞❡ ❘n é ❝❤❛♠❛❞❛

❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❡ é ❞❡♥♦t❛❞❛ ♣♦r B(❘n)✳

❊①❡♠♣❧♦ ✶✳✶✳✾✳ ❈♦♥s✐❞❡r❡♠♦sΩ =❘n, n1✳ ❙❡❥❛ f :n ❘ ♥ã♦ ♥❡❣❛t✐✈❛✱ ✐♥t❡❣rá✈❡❧✱

t❛❧ q✉❡ Rnf(x)dx= 1✳ ❉❡✜♥✐♠♦s

P(A) =

Z

A

f(x)dx, A ∈ B(❘n

).

❊♥tã♦(❘n,

B(❘n),P) é ✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱

P(Ω) = 1 =Z

❘n

f(x)dx.

❙❡ {Ai}i∈◆✱ Ak∩Aj =∅✱ k 6=j ❡♥tã♦

P [

i∈◆

Ai

!

=

Z

S

i∈◆Ai

f(x)dx=

Z

A1∪A2∪...∪An

f(x)dx=X

i∈◆

P(Ai).

❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ q✉❡ ✜③❡♠♦s ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❝♦♥❝❧✉í♠♦s q✉❡ (❘n,

B(❘n),P)é ✉♠

❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✵✳ ❙❡❥❛(Ω,F,P) ✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❆ ❛♣❧✐❝❛çã♦ ❳: Ω→❘n é

❝❤❛♠❛❞❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ✭✈✳❛✳✮n✲❞✐♠❡♥s✐♦♥❛❧ s❡ ♣❛r❛ B ∈ B(❘n)✱ ❳−1(B)∈ F

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✶✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡ ❞✐③❡♠♦s q✉❡ ❳ é F✲♠❡♥s✉rá✈❡❧✳ ❯s✉❛❧♠❡♥t❡ ❡s✲

❝r❡✈❡♠♦s ❳ ❡ ♥ã♦ ❳(w)✳ ❉❡♥♦t❛♠♦s P(❳−1(B)) ♣♦r P(B)

❊①❡♠♣❧♦ ✶✳✶✳✶✷✳ ❙❡❥❛ A∈ F✳ ❊♥tã♦ ❛ ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛ ❞❡ A✱ ❞❛❞❛ ♣♦r✿

✶A(w) =

(

1 s❡ wA

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é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳ ❉❡ ❢❛t♦✱ ♦❜s❡r✈❡♠♦s q✉❡ ♣❛r❛ B ∈ B(❘) t❡♠♦s✿

✶−1

A (B) =

     

    

A, B t❛❧ q✉❡ 1B, 0/ B

Ω, B t❛❧ q✉❡ 1B, 0B AC, B t❛❧ q✉❡ 1/ B, 0B

∅, B t❛❧ q✉❡ 1∈/ B, 0∈/ B

❈♦♠♦ t♦❞❛s ❛s ♣♦ssí✈❡✐s ✐♠❛❣❡♥s ✐♥✈❡rs❛s ❡♥tã♦ ❡♠ F✱ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳

❊①❡♠♣❧♦ ✶✳✶✳✶✸✳ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ❢✉♥çõ❡s ✐♥❞✐❝❛❞♦r❛s ❳ =

n

P

i=1

αi✶Ai✱ ❡♠ q✉❡ αi sã♦ ♥ú♠❡r♦s ❡ Ai ∈ F✱ ❝♦♠

n

S

i=1

Ai = Ω✱ i= 1, . . . , n é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳

▲❡♠❛ ✶✳✶✳✶✹✳ ❙❡❥❛ ❳: Ω❘n ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳ ❊♥tã♦

F(❳) =❳−1(B) :B ∈ B(❘n)

é ✉♠❛ σ✲á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛ ❛ σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r ❳✳ ➱ ❛ ♠❡♥♦r σ✲á❧❣❡❜r❛ ❞❡ F ❝♦♠

r❡s♣❡✐t♦ ❛ q✉❛❧ ❳ é ♠❡♥s✉rá✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ ❳ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱ t❡♠♦s q✉❡ ❳−1() = ∅ ∈ F✳ ▲♦❣♦

∅ ∈ F(❳)✳ ❙❡B ∈ F(❳)❡♥tã♦ ❳−1(B)∈ F ❡ ❝♦♠♦F éσ✲á❧❣❡❜r❛ t❡♠♦s q✉❡ ❳−1(❘n\B) =

Ω\❳−1(B)∈ F✱ ❞♦♥❞❡ ❘n\B ∈ F(❳)✳ ❋✐♥❛❧♠❡♥t❡✱ s❡ (B

k)k1 é ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ F(❳)

❡♥tã♦ ❳−1(B

k)∈ F ♣❛r❛ t♦❞♦ k ≥1 ❡ ♣♦rt❛♥t♦ ❳−1

S

k=1

Bk

= S∞

k=1

❳−1(B

k)∈ F✱ ❥á q✉❡

F é σ✲á❧❣❡❜r❛✳ ▲♦❣♦ S∞

k=1

Bk ∈ F(❳)✳ Pr♦✈❛♠♦s ❡♥tã♦ q✉❡ F(❳) é ✉♠❛σ✲á❧❣❡❜r❛✳

❱❛♠♦s s✉♣♦r q✉❡ G é ✉♠❛ σ✲á❧❣❡❜r❛✱ G ⊂ F t❛❧ q✉❡ ❳ é G✲♠❡♥s✉rá✈❡❧✳ P❡❧❛ ❞❡✜♥✐çã♦

❞♦ F(❳)✱ t❡♠♦s q✉❡ ❳ é F(❳)✲♠❡♥s✉rá✈❡❧✳ ◆♦s r❡st❛ ♣r♦✈❛r q✉❡ F(❳) ⊂ G✳ ❉❛❞♦

❳−1(B)∈ F(❳) ❝♦♠ B ∈ B(❘n)✱ ❝♦♠♦ ❳ é G✲♠❡♥s✉rá✈❡❧✱ ❳−1(B)∈ G✳ ❈♦♠♦ ❳−1(B) ❢♦✐

❛r❜r✐tár✐♦✱ t❡♠♦s q✉❡F(❳)⊂ G✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✺✳ ❙❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❨ é ❢✉♥çã♦ ❞❡ ❳✱ ✐st♦ é✱ s❡ ❨= Φ(❳)♣❛r❛ Φ :❘n

→❘✱ ❡♥tã♦ ❨ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ F(❳)✲♠❡♥s✉rá✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ ❨ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❡♠ ❢✉♥çã♦ ❞❡ ❳✳ ❳: Ω →❘n ✈❛r✐á✈❡❧

❛❧❡❛tór✐❛✳ ▲♦❣♦

❨= Φ(❳)❨= Φ❨: Ω❘.

❉❛❞♦K ∈ B(❘)✱

❨−1(K) = (Φ

◦❳)−1(K) =❳−1 Φ−1(K).

❈♦♠♦ ❳ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱F(❳)✲♠❡♥s✉rá✈❡❧✱

Φ−1(K)∈ B(❘n), −1 Φ−1(K)

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▲♦❣♦ ❨ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛F(❳)✲♠❡♥s✉rá✈❡❧✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✻✳ ❙❡ (Ω,F,P) é ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ ❳ =

n

P

i=1

αi✶Ai é ✉♠❛ ❢✉♥çã♦ s✐♠♣❧❡s✳ ❉❡✜♥✐♠♦s ❛ ✐♥t❡❣r❛❧ ❞❡ ❳ ♣♦r✿

Z

❳(w)dP(w) :=

Z

❳dP:=

n

X

i=1

αiP(Ai).

P❛r❛ ❳ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ♥ã♦ ♥❡❣❛t✐✈❛✱ ❞❡✜♥✐♠♦s

Z

❳(w)dP(w) :=

Z

❳dP:= sup

❨≤❳

Z

❨dP, ❨ ❢✉♥çã♦ s✐♠♣❧❡s.

❊ s❡ ❳: Ω❘ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞❡✜♥✐♠♦s

Z

❳dP:=

Z

❳+dP

Z

❳−dP

❞❡s❞❡ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ✐♥t❡❣r❛✐s ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ s❡❥❛ ✜♥✐t❛✳

◆❡st❡ ❝❛s♦✱ ❳+ = max{,0} ❡ ❳= max{−,0}✱ ❞❡ ♦♥❞❡ ❳=+

P❛r❛ ❳: Ω→❘n✱ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱ ❳= (❳1,2, . . . ,n)✱ ❡s❝r❡✈❡♠♦s

Z

❳dP:=

Z

❳1dP, Z

❳2dP, . . . , Z

❳ndP

.

❉❡✜♥✐çã♦ ✶✳✶✳✶✼✳ ❈❤❛♠❛♠♦s

❊(❳) :=

Z

❳dP

♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ✭♦✉ ✈❛❧♦r ♠é❞✐♦✮ ❞❡ ❳✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✽✳ ❈❤❛♠❛♠♦s

❱(❳) :=

Z

Ω|

❳−❊(❳)|2dP

❛ ✈❛r✐â♥❝✐❛ ❞❡ ❳✳ | · | ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✾✳

❱(❳) = ❊(|❳−❊(❳)|2)

= ❊(||22❳❊(❳) +|❊(❳)|2)

= ❊(|❳|2)2(❊(❳1❊(❳1)) +. . .+❊(❳n❊(❳n

))) +❊(|❊(❳)|2)

(19)

❆✜♥❛❧✱

❊(❳1❊(❳1)) +. . .+❊(❳n(n)) =(1)2+. . .+(n)2 =

|❊(❳)|2 ❡

❊(|❊(❳)|2) = Z

Ω|

❊(❳)|2dP=|()|2 Z

dP =|❊(❳)|2 Z

✶ΩdP=|❊(❳)|2.P(Ω) =|❊(❳)|2.

▲❡♠❛ ✶✳✶✳✷✵ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❤❡❜②s❤❡✈✮✳ ❙❡❥❛ ❳ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❡1≤p≤ ∞✱

❡♥tã♦ P(|❳| ≥λ)≤ 1

λp.❊(|❳|

p) ♣❛r❛ λ >0✳

❉❡♠♦♥str❛çã♦✳

❊(|❳|p

) =

Z

Ω|

❳|p

dP

=

Z

{|❳|≥λ}∪{||}|

❳|pdP

=

Z

{|❳|≥λ}|

❳|p

dP+

Z

{|❳|<λ}|

❳|p

dP

Z

{|❳|≥λ}|

❳|p

dP

=

Z

✶{|❳|≥λ}|❳|pdP.

▼❛s

✶{|❳|≥λ}|❳|p =

(

|❳|p s❡ w∈ {|| ≥λ}

0 ❝❛s♦ ❝♦♥trár✐♦ ❆❧é♠ ❞✐ss♦

✶{|❳|≥λ}λp =

(

λp s❡ w∈ {|| ≥λ}

0 ❝❛s♦ ❝♦♥trár✐♦

❖❜s❡r✈❡ q✉❡ q✉❛♥❞♦w ∈ {|❳| ≥ λ}✱ |❳|p λp ❡ q✉❛♥❞♦ w ∈ {||< λ} ||p =λp✳ ▲♦❣♦

✶{|❳|≥λ}|❳|p ≥✶{||≥λ}λp✳ ❚❡♠♦s ❡♥tã♦ q✉❡

❊(|❳|p

)≥

Z

✶{|❳|≥λ}|❳|pdP=

Z

✶{|❳|≥λ}λpdP=λpP(|❳| ≥λ).

❙❡❥❛ x = (x1, x2, . . . , xn) ❡ y = (y1, y2, . . . , yn) ❡♠ ❘n✳ ❉❡♥♦t❛♠♦s x ≤y q✉❛♥❞♦

xi ≤yi✱ i∈◆✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✶✳ ✭✐✮ ❆ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ❳ é ❛ ❢✉♥çã♦µ❳ :❘n[0,1]❞❡✜♥✐❞❛

(20)

✭✐✐✮ ❙❡ ❳1, . . . ,❳m : Ω → ❘n sã♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s✱ ❞❡✜♥✐♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❥✉♥t❛

❞❡ ❳1, . . . ,❳m ♣♦r µ❳1,...,❳m : (❘n) m

→[0,1] ❞❛❞❛ ♣♦r

µ❳1,...,❳m(z

1, . . . , zm) := P(❳1 ≤z1, . . . ,❳m ≤zm) ❝♦♠ zi ∈❘n, i= 1, . . . , m.

❉❡✜♥✐çã♦ ✶✳✶✳✷✷✳ ❙✉♣♦♥❤❛ q✉❡ ❳ : Ω ❘n é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❢✉♥çã♦ ❞❡

❞✐str✐❜✉✐çã♦ µ❳✳ ❙❡ ❡①✐st✐r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ♥❡❣❛t✐✈❛✱ ✐♥t❡❣rá✈❡❧f :❘n❘ t❛❧ q✉❡✿

µ❳(z) =

z1

Z

−∞

z2

Z

−∞

. . .

zn

Z

−∞

f(x1, . . . , xn)dxn. . . dx1,

❡♥tã♦ f é ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ❳✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✷✸✳ P❛r❛z = (z1, . . . , zn)✱

P(❳≤z) = P(❳−1((−∞, z1]×(−∞, z2]×. . .×(−∞, zn]))

= P(❳((−∞, z1]×(−∞, z2]×. . .×(−∞, zn])

=

z1

Z

−∞

z2

Z

−∞

. . .

zn

Z

−∞

f(x1, . . . , xn)dxn. . . dx1

=

Z

(−∞,z1]×(−∞,z2]×...×(−∞,zn]

f(x)dx

❖❜s❡r✈❛çã♦ ✶✳✶✳✷✹✳ P♦❞❡♠♦s ❞❡✜♥✐r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛B ∈ B(❘n) ❝♦♠♦

P❳(B) =P(❳−1(B)) = P(❳∈B) =

Z

B

f(x)dx.

P❳ é ❝❤❛♠❛❞❛ ❧❡✐ ❞❡ ❳✳

❊①❡♠♣❧♦ ✶✳✶✳✷✺✳ ❙❡❥❛ ❳: Ω❘✱ s❡ ❳ t❡♠ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡

f(x) = √ 1

2πσ2.e −|x−m|2

2σ2 , x∈❘,

❞✐③❡♠♦s q✉❡ ❳ t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ●❛✉ss✐❛♥❛ ✭♦✉ ♥♦r♠❛❧✮✱ ❝♦♠ ♠é❞✐❛ m ❡ ✈❛r✐â♥❝✐❛ σ2✳ ◆❡st❡ ❝❛s♦ ❡s❝r❡✈❡♠♦s ❳ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ N(m, σ2)✳

❊①❡♠♣❧♦ ✶✳✶✳✷✻✳ ❙❡ ❳: Ω→❘n t❡♠ ❞❡♥s✐❞❛❞❡

f(x) = 1

((2π)ndetC)12

.e12(x−m)C−1(x−m), x∈❘n,

♣❛r❛ ❛❧❣✉♠ m ❘n C ♠❛tr✐③ ❞❡✜♥✐❞❛ ♣♦s✐t✐✈❛ ❡ s✐♠étr✐❝❛✱ ❞✐③❡♠♦s q✉❡ ❳ t❡♠ ✉♠❛

(21)

✶✵

❊ ❡s❝r❡✈❡♠♦s ❳ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ N(m, C)✳

▲❡♠❛ ✶✳✶✳✷✼✳ ❙❡❥❛ ❳ : Ω → ❘n ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳ ❆ss✉♠❛ q✉❡ s✉❛ ❢✉♥çã♦ ❞✐str✐✲

❜✉✐çã♦ µ❳ t❡♠ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ f✳ ❙✉♣♦♥❤❛ q✉❡ g : ❘n ❘✱ ❨ = g(❳) é ✐♥t❡❣rá✈❡❧✳

❊♥tã♦

❊(❨) = Z

❘n

g(x)f(x)dx.

❊♠ ♣❛rt✐❝✉❧❛r✱

❊(❳) =

Z

❘n

xf(x)dx ❡ ❱(❳) =

Z

❘n

|x❊(❳)|2f(x)dx.

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s s✉♣♦r q✉❡g é ✉♠❛ ❢✉♥çã♦ s✐♠♣❧❡s ❡♠ ❘n

g =

N

X

i=1

ci✶Bi, Bi ∈ B(❘

n

), i= 1, . . . , N, ci ∈❘.

❊♥tã♦

❊(❨) = ❊(g(❳)) =

Z

N

X

i=1

ciχBi(❳)dP

=

Z

N

X

i=1

ci✶{w:❳(w)∈Bi}(w)dP(w) =

N

X

i=1

ciP(❳∈Bi)

=

N

X

i=1

ci

Z

Bi

f(x)dx

=

N

X

i=1

ci

Z

❘n

✶Bi(x)f(x)dx

=

Z

❘n

N

X

i=1

ci✶Bi(x)f(x)dx

=

Z

❘n

g(x)f(x)dx

❈♦♠♦ t♦❞❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞❛ ♣♦r ❢✉♥çõ❡s s✐♠♣❧❡s✱ ✈❛❧❡ ♣❛r❛ t♦❞❛

g ✐♥t❡❣rá✈❡❧✳

(22)

✶✶

❊(❳) =Z

❳dP=

Z

xf(x)dx

❡ q✉❛♥❞♦ ❳ : Ω ❘n✱ ❛♣❧✐❝❛♠♦s ❡s♣❡r❛♥ç❛ ❡♠ ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛✳ ❆✜♥❛❧✱ ❊( i) =

R

❳idP =

R

❘n

g(x)f(x)dx= R

❘n

xif(x)dx✱ ❡♠ q✉❡f é ❢✉♥çã♦ ❞❡ µ❳✳ ❆ss✐♠✱

❊(❳) =

 Z

❳1dP, . . . , Z

❳ndP

 =   Z ❘n

x1f(x)dx, . . . , Z

❘n

xnf(x)dx

=

Z

❘n

xf(x)dx

P❛r❛ ✈❡r✐✜❝❛r♠♦s ❛ ✈❛r✐â♥❝✐❛✱ ❝♦♥s✐❞❡r❡♠♦sg(x) = |x−❊(❳)|2

❱(❳) =❊ |❊(❳)|2=

Z

❘n

g(x)f(x)dx=

Z

❘n

|x❊(❳)|2f(x)dx.

❊①❡♠♣❧♦ ✶✳✶✳✷✽✳ ❙❡ ❳ é N(m, σ2)✱ ❡♥tã♦ ❊(❳) =R

x.√1

2πσ2.e

−(x−m)2

2σ2 dx=m

❉❡✜♥✐çã♦ ✶✳✶✳✷✾✳ ❉✐③❡♠♦s q✉❡ ♦s ❡✈❡♥t♦s A ❡ B sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s s❡ P(A B) = P(A).P(B)✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✸✵✳ ❙❡ A ❡ B sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❛♠❜é♠ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s AC B

AC BC

❉❡♠♦♥str❛çã♦✳ Pr♦✈❡♠♦s q✉❡AC B sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❈♦♠♦

P(AC) = 1−P(A) ❡B = (A∪AC)∩B = (A∩B)∪(AC ∩B),

t❡♠♦s q✉❡

P(B) =P(A∩B) +P(AC ∩B) = P(A).P(B) +P(AC ∩B).

▲♦❣♦

P(AC ∩B) = P(B)(1−P(A)) = P(B).P(AC).

Pr♦✈❡♠♦s q✉❡AC BC sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❈♦♠♦

(AC

∪A)BC =BC ❡♥tã♦ P(AC

∩BC) +P(A

∩BC) = P(BC).

▼❛s

P(AC

∩BC) +P(A).P(BC) = P(BC)

P(AC

∩BC) =P(BC)

−P(A).P(BC)

⇒P(AC

∩BC) = P(BC) (1

(23)

✶✷

❉❡✜♥✐çã♦ ✶✳✶✳✸✶✳ ❙❡❥❛♠ A1, A2, . . . , Am ❡✈❡♥t♦s✳ ❉✐③❡♠♦s q✉❡ ❡st❡s ❡✈❡♥t♦s sã♦ ✐♥❞❡✲

♣❡♥❞❡♥t❡s s❡ ♣❛r❛ q✉❛❧q✉❡r ❡s❝♦❧❤❛ 1≤k1 < k2 < . . . < km✱ t❡♠♦s

P

m

\

i=1

Aki

!

=P(Ak1 ∩. . .∩Akm) =

m

Y

i=1

P(Aki) = P(Ak1).P(Ak2). . . . .P(Akm).

❉❡✜♥✐çã♦ ✶✳✶✳✸✷✳ ❙❡❥❛♠ Ui ⊆ F σ✲á❧❣❡❜r❛s✱ ♣❛r❛ i ∈ ◆✳ ❉✐③❡♠♦s q✉❡ ❡st❛s σ✲á❧❣❡❜r❛s

Ui✱ i ∈ ◆✱ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s s❡ ♣❛r❛ t♦❞❛ ❡s❝♦❧❤❛ 1 ≤ k1 < k2 < . . . < km ❞❡ ❡✈❡♥t♦s

Aki ∈ Uki✱ t❡♠♦s

P

m

\

i=1

Aki

!

=

m

Y

i=1

P(Aki).

❉❡✜♥✐çã♦ ✶✳✶✳✸✸✳ ❙❡❥❛♠ ❳i : Ω → ❘n✱ i ∈ ◆✱ ❛❧❡❛tór✐❛s✳ ❉✐③❡♠♦s q✉❡ ❛s ✈❛r✐á✈❡✐s

❛❧❡❛tór✐❛s ❳i✱ i∈◆ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s s❡ ❛s σ✲á❧❣❡❜r❛s U(❳i) sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✹✳ ❈♦♥s✐❞❡r❡♠♦s O ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ O ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ O✱ ❛ ❝♦❧❡çã♦ O é ❝❤❛♠❛❞❛ ❞❡ ✉♠ π✲s②st❡♠ s❡ é ❢❡❝❤❛❞♦ s♦❜ ✐♥t❡rs❡çã♦ ✜♥✐t❛✱ ✐st♦ é✱ A, B ∈ O ✐♠♣❧✐❝❛ ❞❡ A B ∈ O✳ ❊st❡ é ✉♠ λ✲s②st❡♠ s❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦

s❛t✐s❢❡✐t❛s✿ ✭✐✮ O ∈ O

✭✐✐✮ A, B ∈ O ❡ AB BA∈ O

✭✐✐✐✮ Ai ∈ O✱ Ai րA✱ i= 1,2, . . . ⇒A∈ O

▲❡♠❛ ✶✳✶✳✸✺✳ ❙❡❥❛l(O) ♦ ♠❡♥♦r λ✲s②st❡♠ q✉❡ ❝♦♥té♠ O✱ ❡♥tã♦ l(O) é σ✲á❧❣❡❜r❛✳

❉❡♠♦♥str❛çã♦✳ ❱❡r ❡♠ ❬✻❪

▲❡♠❛ ✶✳✶✳✸✻✳ ❙❡❥❛♠ O ❡ O˜ ❞✉❛s ❝♦❧❡çõ❡s ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ O ❝♦♠ O ⊂ O˜✳ ❙✉♣♦♥❤❛

O π✲s②st❡♠ ❡ O˜ λ✲s②st❡♠✳ ❊♥tã♦ σ(O)O˜✱ ❡♠ q✉❡ σ(O) é ❛ ♠❡♥♦rσ✲á❧❣❡❜r❛ ❝♦♥t❡♥❞♦

O✳

❉❡♠♦♥str❛çã♦✳ ❖ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞♦ ❧❡♠❛✶✳✶✳✸✺ ♣♦✐s ❥á q✉❡ σ(O) é ❛ ♠❡♥♦r σ✲á❧❣❡❜r❛

❝♦♥t❡♥❞♦O ❡l(O)é ♦ ♠❡♥♦r λ✲s②st❡♠ ❝♦♥t❡♥❞♦O✱ t❡♠♦s σ(O)⊂l(O)⊂O˜✳

▲❡♠❛ ✶✳✶✳✸✼✳ ❙❡❥❛♠ Q, Q′ ❞✉❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡♠ (Ω,F)✳ ❙❡❥❛ C ✉♠ π✲s②st❡♠✱ ❡ F =

σ(C)✳ ❊♥tã♦ Q=Q′ ❡♠ C ✐♠♣❧✐❝❛ Q=Q❡♠ F

❉❡♠♦♥str❛çã♦✳ ■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ✈❡r✐✜❝❛r q✉❡ L := {A ∈ F : Q(A) = Q′(A)} é λ

(24)

✶✸

✭✐✐✮ A∈ L ⇒ AC

∈ L✳ ❉❡ ❢❛t♦✱ ♣♦✐s

Q(AC) =Q(Ω)

−Q(A) = 1Q(A)

Q′(AC) = Q(Ω)Q(A) = 1Q(A).

❈♦♠♦ Q(A) = Q′(A)t❡♠♦s q✉❡ Q(AC) = Q(AC)

✭✐✐✐✮ Ai ∈ L✱Ai րA✱i= 1,2, . . . ⇒A∈ L✳ ❇❛st❛ ✉s❛r♠♦s ❛σ✲❛❞✐t✐✈✐❞❛❞❡ ❡ é ✈❡r❞❛❞❡✐r♦

♣❛r❛ t♦❞❛ ♠❡❞✐❞❛✳

❯s❛♥❞♦ ❛❣♦r❛ ♦ ❧❡♠❛ ✶✳✶✳✸✻✱ ❝♦♠♦ L é λ✲s②st❡♠ q✉❡ ❝♦♥té♠ ♦ π✲s②st❡♠ C ❡♥tã♦ t❡♠♦s

q✉❡Q(B) = Q′(B) B σ(C) =F

▲❡♠❛ ✶✳✶✳✸✽✳ ❙❡❥❛♠ ❳1,❳2, . . .✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✱(i1, i2, . . .), (j1, j2, . . .)❝♦♥❥✉♥t♦s

❞✐s❥✉♥t♦s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❊♥tã♦ F1 = σ(❳i1,❳i2, . . .)✱ F2 = σ(❳j1,❳j2, . . .) sã♦

✐♥❞❡♣❡♥❞❡♥t❡s✳

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦D∈ F2 ❞❡ ❢♦r♠❛ q✉❡

D ={❳j1 ∈B1, . . . ,❳jm ∈Bm}, Bk ∈ B1, k= 1, . . . , m. ❉❡✜♥❛ ❞✉❛s ♠❡❞✐❞❛sQ1 ❡ Q′1 ❡♠ F1✱ ♣❛r❛ A∈ F1✱

Q1(A) = P(A∈D), Q1′(A) =P(A)P(D).

❈♦♥s✐❞❡r❡ ❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉♥t♦s C ⊂ F1 ❞❛ ❢♦r♠❛

C={❳i1 ∈E1, . . . ,❳in ∈En}, El ∈ B1, l = 1, . . . , n. ❡♠ q✉❡ B1 é ❛ σ✲❛❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ [0,1])✳ ◆♦t❡ q✉❡

Q1(C) = P

n

\

l=1

{❳il ∈El}

m

\

k=1

{❳jk ∈Bk}

!

=

n

Y

l=1

P(❳il ∈El)

m

Y

k=1

P(❳jk ∈Bk) = P(C)P(D) =Q′1(C)

❊♥tã♦ Q1 = Q′1 ❡♠ C✱ C é ❢❡❝❤❛❞♦ s♦❜ ✐♥t❡rs❡çõ❡s✱ σ(C) = F1 ⇒ Q1 = Q′1 ❡♠ F1✳ ✭✈❡r

❧❡♠❛ ✶✳✶✳✸✼✮✳

❘❡♣❡t✐♥❞♦ ♦ ❛r❣✉♠❡♥t♦ ✜①❛♠♦sA ∈ F1 ❡ ❞❡✜♥✐♠♦sQ2, Q′2 ❡♠F2 ♣♦r P(A∩·), P(A)P(·)✳

P❡❧♦ ♣r❡❝❡❞❡♥t❡✱ ♣❛r❛ q✉❛❧q✉❡rD✱ Q2(D) =Q′2(D)✱ ✐♠♣❧✐❝❛Q2 =Q′2 ❡♠ F2 ❡ ❡♥tã♦ ♣❛r❛

(25)

✶✹

▲❡♠❛ ✶✳✶✳✸✾✳ ❙❡❥❛♠ ❳1, . . . ,❳m+n ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛❧♦r❡s ❡♠

❘d✳ ❙✉♣♦♥❤❛ q✉❡ f : (❘d)n ❘ s❡❥❛ ♠❡♥s✉rá✈❡❧ ❝♦♠ r❡❧❛çã♦ à σ✲á❧❣❡❜r❛ B((❘d)n)

g : (❘d)m ❘ s❡❥❛ ♠❡♥s✉rá✈❡❧ ❝♦♠ r❡❧❛çã♦ àσ✲á❧❣❡❜r❛B((❘d)m)✱ ❡♥tã♦ ❨=f(❳

1, . . . ,❳n)

❡ ❩=g(❳n+1, . . . ,❳m+n) sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳

❉❡♠♦♥str❛çã♦✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❨ ❡ ❩ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳ P❛r❛ ✐ss♦ ❜❛st❛ ♠♦str❛r✲ ♠♦s q✉❡σ(❨)é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ σ(❩)✱ ♦✉ s❡❥❛✱

P[❨∈B1, ❩∈B2] =P[❨∈B1]P[❩∈B2] ∀B1, B2.

▼❛s ❛σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ❞♦ ❨ é ✉♠❛ s✉❜✲σ✲á❧❣❡❜r❛ ❞❛ σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r (❳1, . . . ,❳n)✱

❡ s✐♠✐❧❛r♠❡♥t❡ ♣❛r❛ σ(❩) ❡ σ(❳n+1, . . . ,❳m+n)✳ ❉❡ ❢❛t♦✱ ♥♦t❡♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r

❝♦♥❥✉♥t♦ B ∈ B(❘)✱ t❡♠♦s

❨−1(B) = (f (❳

1, . . . ,❳n))−1(B)

= (❳1, . . . ,❳n)−1

 

 f

−1(B) | {z }

∈ B((❘d)n), ♣♦✐s f é B((d)n)✲♠❡♥s✉rá✈❡❧

  

= (❳1, . . . ,❳n)−1(❛❧❣✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❇♦r❡❧)∈σ(❳1, . . . ,❳n)

P❡❧♦ ❧❡♠❛ ✶✳✶✳✸✽✱σ(❳1, . . . ,❳n)é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ σ(❳n+1, . . . ,❳n+m)✳ ▲♦❣♦

σ(❨)⊂σ(❳1, . . . ,❳n) é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ σ(❩)⊂σ(❳n+1, . . . ,❳n+m)

❚❡♦r❡♠❛ ✶✳✶✳✹✵✳ ❙❡❥❛♠ ❳1, . . . ,❳m : Ω → ❘n ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s✳ ❊st❛s ✈❛r✐á✈❡✐s

❛❧❡❛tór✐❛s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s s❡ ❡ s♦♠❡♥t❡ s❡ ✭✐✮ µ❳1,...,❳m(x

1, . . . , xm) = µ❳1(x1). . . µ❳m(xm) ♣❛r❛ t♦❞♦s x1, . . . , xm ∈❘n✳

❙❡ ❛s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ♣♦ss✉❡♠ ❢✉♥çõ❡s ❞❡ ❞❡♥s✐❞❛❞❡s r❡s♣❡❝t✐✈❛s f❳1, . . . , fm✱ ✭✐✮ é ❡q✉✐✈❛❧❡♥t❡ ❛

✭✐✐✮ f❳1,...,❳m(x1, . . . , xm) = f❳1(x1). . . f❳m(xm) ♣❛r❛ t♦❞♦s x1, . . . , xm ∈❘

n

❉❡♠♦♥str❛çã♦✳ ❱❡r ❡♠ ❬✼❪

❚❡♦r❡♠❛ ✶✳✶✳✹✶✳ ❙✉♣♦♥❤❛ q✉❡ ❳1, . . . ,❳m : Ω→❘ sã♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥✲

t❡s ❝♦♠

(26)

✶✺

❡♥tã♦ ❊(|❳1❳2. . .❳m|)<∞ ❡

❊(❳1❳2. . .❳m) = ❊(❳1). . .❊(Xm).

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ❩1 =✶A1, . . . ,❩m=✶Am ∈ F✳ ❈❛❧❝✉❧❡♠♦s

❊(|❩1.· · · .❩m|) =

Z

|✶A1(w).· · · .✶Am(w)| dP =

Z

✶A1∩...∩Am(w)dP = P(A1∩. . .∩Am)<∞

❈♦♥s✐❞❡r❡♠♦sA1 =❩−11(D1), . . . , Am =❩m−1(Dm) ❝♦♠ D1, . . . , Dm ∈ B(❘)✳

❈♦♠♦

❊(❩1❩2.· · · .❩m) =

Z

✶A1(w).· · · .✶Am(w)dP =

Z

✶A1∩...∩Am(w)dP = P(A1∩. . .∩Am)

= P(A1).· · · .P(Am)

= ❊(❩1).· · · .❊(❩m)

❈♦♠♣❧❡t❛♥❞♦ ❛ ♣r♦✈❛ ♣❛r❛ ❢✉♥çõ❡s ✐♥❞✐❝❛❞♦r❛s✳ ❖ q✉❡ ❜❛st❛ ♣❛r❛ ❝♦♠♣❧❡t❛r♠♦s ❛ ♣r♦✈❛✱ ✉♠❛ ✈❡③ q✉❡✱ ❛s ❢✉♥çõ❡s s✐♠♣❧❡s sã♦ ❡s❝r✐t❛s ❝♦♠♦ s♦♠❛ ❞❡ ❢✉♥çõ❡s ✐♥❞✐❝❛❞♦r❛s ❡ sã♦ ❞❡♥s❛s ♥♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ✐♥t❡❣rá✈❡✐s✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✷✳ ❙❡❥❛♠ A1, A2, . . . ❡✈❡♥t♦s ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❊♥tã♦ ♦

❡✈❡♥t♦ lim supAn :=

T

k=1

S

n=k

An = {ω ∈ Ω : ω ∈ An ♣❛r❛ ✐♥✜♥✐t♦s ✈❛❧♦r❡s ❞❡ ♥ } é

❝❤❛♠❛❞♦ An i−o✳

▲❡♠❛ ✶✳✶✳✹✸ ✭❇♦r❡❧ ❈❛♥t❡❧❧✐✮✳ ❙❡ P∞

n=1P(

An)<∞✱ ❡♥tã♦ P(An i−o) = 0✳

❉❡♠♦♥str❛çã♦✳ P(An i−o) = P

T

k=1

S

n=k

An

≤ P

S

n=k

An

≤ P∞

n=k

P(An) ց 0✱ q✉❛♥❞♦

k→ ∞✳ ▲♦❣♦ P(An i−o) = 0✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✹✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s {❳k}k ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠❛

✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❳ ❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ s❡ ❞❛❞♦ ǫ✱ lim

k→∞P(|❳k−❳|< ǫ) = 1✳

❚❡♦r❡♠❛ ✶✳✶✳✹✺✳ ❙❡ ❳k →❳ ❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛

❳kj

j=1 ⊂

{❳k}∞k=1 t❛❧ q✉❡

(27)

✶✻

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❝❛❞❛ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦j✱ ✈❛♠♦s t♦♠❛rkj ❣r❛♥❞❡ t❛❧ q✉❡

P

❳kj −❳

> 1

j

≤ 1

j2

❡ t❛♠❜é♠ . . . < kj−1 < kj < . . . , kj → ∞✳ ❙❡❥❛ Aj :=

n

kj−❳> 1j o

✳ ❏á q✉❡

P

j=1 1

j2 <∞✱ ♦ ❧❡♠❛ ❞❡ ❇♦r❡❧ ❈❛♥t❡❧❧✐ ✐♠♣❧✐❝❛ q✉❡ P(Aj ✐✲♦) = 0✳ P♦rt❛♥t♦ ♣❛r❛ q✉❛s❡ t♦❞♦

♣♦♥t♦ w✱ ❳kj(w)−❳(w)

≤ 1j ❢♦r♥❡❝✐❞♦ j ≥ J✱ ♣❛r❛ ❛❧❣✉♠ J ✐♥❞❡①❛❞♦ ❞❡♣❡♥❞❡♥❞♦ ❞❡

w✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✻✳ ❙❡❥❛ ❳ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ t❛❧ q✉❡ ❳: Ω❘n✳ ❊♥tã♦

φ❳(λ) := ❊(eiλ❳), λ∈❘n,

é ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❳✳

▲❡♠❛ ✶✳✶✳✹✼✳ ✭✐✮ ❙❡ ❳1, . . . ,❳m sã♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛

λ ∈❘n✱ ❡♥tã♦

φ❳1+...+❳m(λ) = φ❳1(λ).· · · .φ❳m(λ). ✭✐✐✮ ❙❡ ❳ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱ ❳: Ω→❘✱ ❡♥tã♦

φ(k)(0) =ik❊(❳k), k ∈◆.

✭✐✐✐✮ ❙❡ ❳ ❡ ❨ sã♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❡φ(λ) =φ(λ) ♣❛r❛ t♦❞♦λ✱ ❡♥tã♦

µ❳(x) = µ❨(x), ♣❛r❛ t♦❞♦ x.

❉❡♠♦♥str❛çã♦✳ ✭✐✮

φ❳1+...+❳m(λ) = ❊ e

iλ(❳1,...,❳m)

= ❊ eiλ❳1, eiλ❳2, . . . , eiλ❳m = ❊ eiλ❳1.

· · · .❊ eiλ❳m ♣❡❧❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ = φ❳1(λ).· · · .φm(λ)

✭✐✐✮ ❚❡♠♦s q✉❡φ′(λ) =i❊ ❳.e ❡ ❡♥tã♦ φ(0) =i❊(❳)✳

P❛r❛ k = 2✱ t❡♠♦s q✉❡ φ2(λ) =i2❊ ❳2e ❡ ❡♥tã♦ φ2(0) =i2❊(❳2)✳

❙✉♣♦♥❤❛ ✈á❧✐❞♦ ♣❛r❛ n✳ ▲♦❣♦

φn+1(λ) = (φn(λ))′ = in❊ ❳neiλ❳′

(28)

✶✼

❡ ❡♥tã♦ φn+1=in+1❊(❳n+1)✳

✭✐✐✐✮ ❱❡r ❬✷❪ ♣❛r❛ ♣r♦✈❛✳

▲❡♠❛ ✶✳✶✳✹✽ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●r♦♥✇❛❧❧✮✳ ❙✉♣♦♥❤❛ q✉❡h∈L1([t, T];❘)α L([t, T];❘)

s❛t✐s❢❛③❡♥❞♦✱ ♣❛r❛ ❛❧❣✉♠β 0✱

0≤h(s)≤α(s) +β

Z s

t

h(λ)dλ ♣❛r❛ s∈[t, T]. ✭✶✳✶✮

❊♥tã♦

h(s)≤α(s) +β

Z s

t

α(λ)e−β(λ−s)dλ ♣❛r❛ s∈[t, T].

❙❡ ❡♠ ❛❞✐çã♦✱α é ❝r❡s❝❡♥t❡✱ ❡♥tã♦ h(s)≤α(s)e−β(st) ♣❛r❛ s[t, T]✳

❉❡♠♦♥str❛çã♦✳ ❆ss✉♠❛ q✉❡β 6= 0✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♦ ❧❡♠❛ é tr✐✈✐❛❧✳

❉❡✜♥❛z(s) :=e−β(s−t)Rs

t h(λ)dλ✳ ❊♥tã♦

z′(s) = βe−β(s−t) Z s

t

h(λ)dλ+e−β(s−t).h(s)

= e−β(s−t)

−β

Z s

t

h(λ)dλ+h(s)

≤ e−β(s−t)α(s) ♣❛r❛s

∈[t, T].

■♥t❡❣r❛♥❞♦ ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s✱

z(s)

Z s

t

e−β(λ−t)α(λ)

⇒e−β(s−t) Z s

t

h(λ)dλ

Z s

t

e−β(λ−t)α(λ)dλ.

▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r eβ(s−t) Z s

t

h(λ)dλ eβs

Z s

t

α(λ)e−βλ ♣❛r❛ s

∈[t, T].

(29)

✶✽

h(s) α(s) +β

Z s

t

α(λ)e−β(λ−s)

≤ α(s) +βα(s)

Z s

t

e−β(λ−s)

= α(s) +α(s) −e−β(λ−s)|s t

= α(s) +α(s) 1 +e−β(t−s)

= α(s)e−β(t−s).

### ✶✳✷ ❊s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧

◆❡st❛ s❡ssã♦ ✈❛♠♦s ❡①♣❧♦r❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧✱ ❛ q✉❛❧ s❡rá ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ♣ró①✐♠❛ s❡ssã♦ ❡ ♦✉tr♦s✳ ❙❡❥❛(Ω,F,P)❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✜①❛❞♦✳ P❛r❛1p <✱ ✐r❡♠♦s ✉s❛rLp(Ω) ♣❛r❛ ❞❡♥♦t❛r ♦ ❡s♣❛ç♦ ❞❡ t♦❞❛s ❛s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s

❳ ❝♦♠ ❊(||p)<✳ ❊st❡ é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ♥♦r♠❛

||❳||p = (❊(|❳|p))

1

p.

◆❡st❛ s❡çã♦ ✐r❡♠♦s ✉s❛r ♦ ❡s♣❛ç♦ L1(Ω)✳ ❊♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s ❡s❝r❡✈❡r❡♠♦s L1(Ω,F)

♣❛r❛ ❡♥❢❛t✐③❛r ❛ σ✲á❧❣❡❜r❛ F

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛ ❳∈L1(Ω,F)✳ ❙✉♣♦♥❤❛ q✉❡ t❡♠♦s ✉♠❛ ♦✉tr❛ σ✲á❧❣❡❜r❛ G ⊂ F✳ ❆

❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ ❳ ❞❛❞♦ G é ❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠❛ ú♥✐❝❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❨ ✭s♦❜

❛ ♠❡❞✐❞❛ P✮ s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ✭✐✮ ❨ é G✲♠❡♥s✉rá✈❡❧❀

✭✐✐✮ R

A

❳dP=R

A

❨dP ♣❛r❛ t♦❞♦ A∈ G

❯s❛r❡♠♦s ❧✐✈r❡♠❡♥t❡ ❊(❳|G) ♣❛r❛ ❞❡♥♦t❛r ❛ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ ❳ ❞❛❞♦

G✳ ◆♦t❡ q✉❡ ❛ G✲♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ♥❛ ❝♦♥❞✐çã♦ ✭✐✮ é ❝r✉❝✐❛❧✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♣♦❞❡rí❛♠♦s

t♦♠❛r ❨=❳ ♣❛r❛ s❛t✐s❢❛③❡r ❛ ❝♦♥❞✐çã♦ ✭✐✐✮✱ ❡ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ♥ã♦ s❡r✐❛ tã♦ s✐❣♥✐✜❝❛t✐✈❛✳ ❆ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ❊(❳|G) ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ❛ ♠❡❧❤♦r ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦ ✈❛❧♦r ❞❡ ❳ ❜❛s❡❛❞❛ ♥❛s ✐♥❢♦r♠❛çõ❡s ♣r♦✈❡♥✐❡♥t❡s ❞❡G✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❙✉♣♦♥❤❛ G = {∅,Ω}✳ ❙❡❥❛ ❳ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❡♠ L1(Ω) ❡ s❡❥❛

❨=❊(❳|G)✳ ❏á q✉❡ ❨ éG✲♠❡♥s✉rá✈❡❧✱ ❡❧❛ ❞❡✈❡ s❡r ✉♠❛ ❝♦♥st❛♥t❡✱ ❞✐r❡♠♦s ❨=c✳ ❊♥tã♦

❜❛st❛ ✉s❛r♠♦s ❛ ❝♦♥❞✐çã♦ ✭✐✐✮ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ❝♦♠ A= Ω ♣❛r❛ t❡r♠♦s✿

Z

❳dP=

Z

(30)

✶✾

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ c = ❊(❳) ❡ ♥ós t❡♠♦s ❊(❳|G) = ❊(❳)✳ ❊ss❛ ❝♦♥❝❧✉sã♦ é ✐♥t✉✐t✐✈❛✲ ♠❡♥t❡ ó❜✈✐❛ ❥á q✉❡ ❛σ✲á❧❣❡❜r❛ G ♥❛♦ ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çã♦✳

❚❡♦r❡♠❛ ✶✳✷✳✸✳ ❙❡❥❛ ❳ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ✐♥t❡❣rá✈❡❧✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ σ✲á❧❣❡❜r❛

G ⊂ F✱ ❛ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ❊(❳|G) ❡①✐st❡ ❡ é ú♥✐❝❛ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ G✲♠❡♥s✉rá✈❡❧

❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ③❡r♦✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❡♠ ❬✼❪

❖❜s❡r✈❡ q✉❡ ❛ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ❡s♣❡r❛♥ç❛ é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❆❜❛✐①♦ ✐r❡♠♦s ❧✐st❛r ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❛ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧✳

✶✳ ❊(❊(❳|G)) =❊(❳)✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ♣r♦✈❛r♠♦s ❜❛st❛ t♦♠❛r♠♦s A = Ω ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧✳

✷✳ ❙❡ ❳ é G✲♠❡♥s✉rá✈❡❧✱ ❡♥tã♦ ❊(❳|G) = ❳✳

❉❡♠♦♥str❛çã♦✳ ❙❡ ❳ é G✲♠❡♥s✉rá✈❡❧ ❡♥tã♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦✱ é ✉♠❛ ✈❡rsã♦ ❞❛ ❡s♣❡✲

r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ ❳ ❞❛❞♦ G✳

✸✳ ❙❡ a, bsã♦ ❝♦♥st❛♥t❡s✱ ❊(a❳+b❩|G) =a❊(❳|G) +b❊(❩|G)✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ t♦❞♦A ∈ G✱ s❡❣✉❡ ❞❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡ ❡ ❞❛

❞❡✜♥✐çã♦ ❞❡ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ q✉❡

Z

A

❊(a❳+b❩|G)dP =

Z

A

a❳+b❩dP

= a

Z

A

❳dP+b

Z

A

❩dP

= a

Z

A

❊(X|G)dP+b

Z

A

❊(❩|G)dP

▲♦❣♦ ❊(a❳+b❩|G) =a❊(❳|G) +b❊(❩|G) P✲q✉❛s❡ s❡♠♣r❡✳

(31)

✷✵

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ ❳ = ✶B ♣❛r❛ ❛❧❣✉♠ B ∈ G✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ A∈ G

t❡♠♦s q✉❡

Z

A

❊(✶B❩|G)dP =

Z

A

✶B❩dP

=

Z

A∩B

❩dP

=

Z

A∩B

❊(Z|G)dP

=

Z

A

✶B❊(❩|G)dP.

▲♦❣♦ ❊(✶B❩|G) =✶B❊(❩|G)✳ ❆♣r♦①✐♠❛♥❞♦ ❳ ♣♦r ❢✉♥çõ❡s s✐♠♣❧❡s ❡ ✉s❛♥❞♦ ♦ t❡♦✲

r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛ ♦❜té♠✲s❡ ♦ r❡s✉❧t❛❞♦✳ ✺✳ ❙❡ ❳ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ G✱ ❡♥tã♦ ❊(❳|G) = ❊(❳)✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ ❳ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡G❡♥tã♦ σ(❳)é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ G✳ ▲♦❣♦

❛s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✶A ❡ ❳ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ♣❛r❛ t♦❞♦ A ∈ G✳ ❙❡❣✉❡ q✉❡

Z

A

❊(❳|G)dP=❊(✶A.❳) = ❊(✶A)❊(❳) =

Z

A

❊(❳)dP.

❈♦♠♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ A∈ G t❡♠♦s q✉❡ ❊(❳|G) = ❊(❳)✳ ✻✳ ❙❡ H ⊆ G ❡♥tã♦ ❊(❳|H) =❊(❊(❳|G)|H)✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ H ⊆ G t❡♠♦s q✉❡

Z

B

❊(❳|H)dP=

Z

B

❳dP=

Z

B

❊(❳|G)dP, B ∈ H.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ❊(❳|G) é ✉♠❛ ❢✉♥çã♦ G✲♠❡♥s✉rá✈❡❧✱ ❛ s✉❛ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐✲ ♦♥❛❧ ❞❛❞♦ H s❛t✐s❢❛③✱

Z

B

❊(❊(❳|G)|H)dP=

Z

B

❊(❳|G)dP, ∀B ∈ H.