❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■▼
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦
❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❍❡r❡❞✐tár✐❛s ❊st♦❝ást✐❝❛s ❡
♦ Pr♦❜❧❡♠❛ ❞❡ P♦rt❢ó❧✐♦ Ót✐♠♦
▼❛r✐❛♥❛ ❙✐❧✈❛ ❚❛✈❛r❡s
❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❍❡r❡❞✐tár✐❛s ❊st♦❝ást✐❝❛s ❡
♦ Pr♦❜❧❡♠❛ ❞❡ P♦rt❢ó❧✐♦ Ót✐♠♦
▼❛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❛♥✳
❚❛✈❛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✉❧♦✳
❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❍❡r❡❞✐tár✐❛s ❊st♦❝ást✐❝❛s ❡
♦ Pr♦❜❧❡♠❛ ❞❡ P♦rt❢ó❧✐♦ Ót✐♠♦
▼❛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❛ ❙✐❧✈❛
❆❣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❡♥❝✐❛❧✳
✏➱ ♠❡❧❤♦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❛✑
❘❡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❛✳
❆❜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✳
❙✉♠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♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
■♥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❡♠♣♦ t≥0✳
❚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✳
❈❛♣ít✉❧♦ ✶
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♦❜❛❜✐❧✐✲
✹
✭✐✮ 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 w∈Ωsã♦ ❝❤❛♠❛❞♦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
✺
❉❛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, n≥1✳ ❙❡❥❛ f :❘n →❘ ♥ã♦ ♥❡❣❛t✐✈❛✱ ✐♥t❡❣rá✈❡❧✱
t❛❧ q✉❡ R❘nf(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❡ w∈A
✻
é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳ ❉❡ ❢❛t♦✱ ♦❜s❡r✈❡♠♦s q✉❡ ♣❛r❛ B ∈ B(❘) t❡♠♦s✿
✶−1
A (B) =
A, B t❛❧ q✉❡ 1∈B, 0∈/ B
Ω, B t❛❧ q✉❡ 1∈B, 0∈B AC, B t❛❧ q✉❡ 1∈/ B, 0∈B
∅, 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)k≥1 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)
✼
▲♦❣♦ ❨ 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
Ω
❳(w)dP(w) :=
Z
Ω
❳dP:=
n
X
i=1
αiP(Ai).
P❛r❛ ❳ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ♥ã♦ ♥❡❣❛t✐✈❛✱ ❞❡✜♥✐♠♦s
Z
Ω
❳(w)dP(w) :=
Z
Ω
❳dP:= sup
❨≤❳
Z
Ω
❨dP, ❨ ❢✉♥çã♦ s✐♠♣❧❡s.
❊ s❡ ❳: Ω→❘ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞❡✜♥✐♠♦s
Z
Ω
❳dP:=
Z
Ω
❳+dP
−
Z
Ω
❳−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
Ω
❳dP:=
Z
Ω
❳1dP, Z
Ω
❳2dP, . . . , Z
Ω
❳ndP
.
❉❡✜♥✐çã♦ ✶✳✶✳✶✼✳ ❈❤❛♠❛♠♦s
❊(❳) :=
Z
Ω
❳dP
♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ✭♦✉ ✈❛❧♦r ♠é❞✐♦✮ ❞❡ ❳✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✽✳ ❈❤❛♠❛♠♦s
❱(❳) :=
Z
Ω|
❳−❊(❳)|2dP
❛ ✈❛r✐â♥❝✐❛ ❞❡ ❳✳ | · | ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛✳
❖❜s❡r✈❛çã♦ ✶✳✶✳✶✾✳
❱(❳) = ❊(|❳−❊(❳)|2)
= ❊(|❳|2−2❳❊(❳) +|❊(❳)|2)
= ❊(|❳|2)−2(❊(❳1❊(❳1)) +. . .+❊(❳n❊(❳n
))) +❊(|❊(❳)|2)
✽
❆✜♥❛❧✱
❊(❳1❊(❳1)) +. . .+❊(❳n❊(❳n)) =❊(❳1)2+. . .+❊(❳n)2 =
|❊(❳)|2 ❡
❊(|❊(❳)|2) = Z
Ω|
❊(❳)|2dP=|❊(❳)|2 Z
Ω
dP =|❊(❳)|2 Z
Ω
✶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}∪{|❳|<λ}|
❳|pdP
=
Z
{|❳|≥λ}|
❳|p
dP+
Z
{|❳|<λ}|
❳|p
dP
≥
Z
{|❳|≥λ}|
❳|p
dP
=
Z
Ω
✶{|❳|≥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
Ω
✶{|❳|≥λ}|❳|pdP=
Z
Ω
✶{|❳|≥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]❞❡✜♥✐❞❛
✾
✭✐✐✮ ❙❡ ❳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❡♠ ✉♠❛
✶✵
❊ ❡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
Ω
N
X
i=1
ciχBi(❳)dP
=
Z
Ω
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✈❡❧✳
✶✶
❊(❳) =Z
Ω
❳dP=
Z
❘
xf(x)dx
❡ q✉❛♥❞♦ ❳ : Ω → ❘n✱ ❛♣❧✐❝❛♠♦s ❡s♣❡r❛♥ç❛ ❡♠ ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛✳ ❆✜♥❛❧✱ ❊(❳ i) =
R
Ω
❳idP =
R
❘n
g(x)f(x)dx= R
❘n
xif(x)dx✱ ❡♠ q✉❡f é ❢✉♥çã♦ ❞❡ µ❳✳ ❆ss✐♠✱
❊(❳) =
Z
Ω
❳1dP, . . . , Z
Ω
❳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
✶✷
❉❡✜♥✐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 ❡ A⊂B ⇒B−A∈ 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✲
✶✸
✭✐✐✮ A∈ L ⇒ AC
∈ L✳ ❉❡ ❢❛t♦✱ ♣♦✐s
Q(AC) =Q(Ω)
−Q(A) = 1−Q(A)
Q′(AC) = Q′(Ω)−Q′(A) = 1−Q′(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❛
✶✹
▲❡♠❛ ✶✳✶✳✸✾✳ ❙❡❥❛♠ ❳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, . . . , f❳m✱ ✭✐✮ 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 ❝♦♠
✶✺
❡♥tã♦ ❊(|❳1❳2. . .❳m|)<∞ ❡
❊(❳1❳2. . .❳m) = ❊(❳1). . .❊(Xm).
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ❩1 =✶A1, . . . ,❩m=✶Am ∈ F✳ ❈❛❧❝✉❧❡♠♦s
❊(|❩1.· · · .❩m|) =
Z
Ω
|✶A1(w).· · · .✶Am(w)| dP =
Z
Ω
✶A1∩...∩Am(w)dP = P(A1∩. . .∩Am)<∞
❈♦♥s✐❞❡r❡♠♦sA1 =❩−11(D1), . . . , Am =❩m−1(Dm) ❝♦♠ D1, . . . , Dm ∈ B(❘)✳
❈♦♠♦
❊(❩1❩2.· · · .❩m) =
Z
Ω
✶A1(w).· · · .✶Am(w)dP =
Z
Ω
✶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✉❡
✶✻
❉❡♠♦♥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).· · · .φ❳m(λ)
✭✐✐✮ ❚❡♠♦s q✉❡φ′(λ) =i❊ ❳.eiλ❳ ❡ ❡♥tã♦ φ′(0) =i❊(❳)✳
P❛r❛ k = 2✱ t❡♠♦s q✉❡ φ2(λ) =i2❊ ❳2eiλ❳ ❡ ❡♥tã♦ φ2(0) =i2❊(❳2)✳
❙✉♣♦♥❤❛ ✈á❧✐❞♦ ♣❛r❛ n✳ ▲♦❣♦
φn+1(λ) = (φn(λ))′ = in❊ ❳neiλ❳′
✶✼
❡ ❡♥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(s−t) ♣❛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)dλ
⇒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−βλdλ ♣❛r❛ s
∈[t, T].
✶✽
h(s) ≤ α(s) +β
Z s
t
α(λ)e−β(λ−s)dλ
≤ α(s) +βα(s)
Z s
t
e−β(λ−s)dλ
= 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❛1≤p <∞✱ ✐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
Ω
❳dP=
Z
Ω
✶✾
❈♦♥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❡✳
✷✵
❉❡♠♦♥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.