JOÃO FILIPE LITA DA SILVA

❏❖➹❖ ❋■▲■P❊ ▲■❚❆ ❉❆ ❙■▲❱❆

❈❖◆❙■❙❚✃◆❈■❆ ❋❖❘❚❊ ❉❊ ❊❙❚■▼❆❉❖❘❊❙✿

❖ ❊❙❚■▼❆❉❖❘ ❉❖❙ ▼✁■◆■▼❖❙ ◗❯❆❉❘❆❉❖❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ●r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠✁❛t✐❝❛✱ ♥❛ ❡s♣❡❝✐❛✲ ❧✐❞❛❞❡ ❞❡ ❊st❛t✁✙st✐❝❛✱ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ◆♦✈❛ ❞❡ ▲✐s❜♦❛✱ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛✳

❖❘■❊◆❚❆❉❖❘✿ P❘❖❋❊❙❙❖❘ ❉❖❯❚❖❘ ❏❖➹❖ ❚■❆●❖ ▼❊❳■❆ ❈❖✲❖❘■❊◆❚❆❉❖❘✿ P❘❖❋❊❙❙❖❘ ❉❖❯❚❖❘ P❊❉❘❖ ❈❖❘❚❊ ❘❊❆▲

■❢ ✇❡ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥ ❆♥❞ ✐❢ ✇❡ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥✱ t✉r♥

❚❤❡♥ ✇❡ ♠✐❣❤t ❧❡❛r♥✱ ❧❡❛r♥

❋r❛♥❝✐s ❍❡❛❧②

❆❣r❛❞❡❝✐♠❡♥t♦s

❖s ♠❡✉s ♣r✐♠❡✐r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ✈ã♦ ♥❛t✉r❛❧♠❡♥t❡ ♣❛r❛ ♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❏♦ã♦ ❚✐❛❣♦ ▼❡①✐❛ q✉❡✱ ❞❡s❞❡ ♦ ♣r✐♠❡✐r♦ ❞✐❛✱ ❛❝r❡❞✐t♦✉ s❡r ♣♦ss✁✙✈❡❧ ❛ ❝♦♥❝r❡t✐③❛çã♦ ❞❡st❡ ♠❡✉ s♦♥❤♦✱ ♥ã♦ ❝♦❧♦❝❛♥❞♦ q✉❛✐sq✉❡r ♦❜st✁❛❝✉❧♦s ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❜❛s❡ ❡♠ ❛♥✁❛❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❡ ❡q✉❛çõ❡s ❝♦♠ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s✱ ♠❛s ❡♥❝❛r❛♥❞♦ ✐ss♦ ❝♦♠♦ ✉♠❛ ❡✈❡♥t✉❛❧ ✈❛♥t❛❣❡♠✳ ❉❡✈♦ ❞✐③❡r q✉❡ ❡st❡ ♠❛❣♥✁✙✜❝♦ ❡ ❡♥t✉s✐❛s♠❛♥t❡ ❞❡s❛✜♦ ♠❡ ❞❡✉ ✉♠ ♥♦✈♦ ✂❛♥✐♠♦ r❡❧❛t✐✈❛♠❡♥t❡ ❛ ♠❛t❡♠✁❛t✐❝❛✱ ❛♦ t❡r ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡st✉❞❛r ❡s✲ t❛t✁✙st✐❝❛ ❡ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ✁❛r❡❛s q✉❡ ♣❛r❛ ♠✐♠ ❝♦♥st✐t✉✐❛♠ ✉♠ ♠✉♥❞♦ ❛ ❞❡s❝♦❜r✐r✳ ❊ ♥❡st❡ ♣♦♥t♦ ♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❏♦ã♦ ❚✐❛❣♦ ▼❡①✐❛ t❡✈❡ ✉♠ ♣❛♣❡❧ ❛❜s♦❧✉t❛♠❡♥t❡ ❢✉❧❝r❛❧ ❛♦ ♣❛rt✐❧❤❛r ❝♦♠✐❣♦ ♦ s❡✉ ✈❛st✁✙ss✐♠♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❛ s✉❛ ❡①♣❡r✐ê♥❝✐❛ ❞❡ ♠✉✐t♦s ❛♥♦s ♥❡st❛s ✁❛r❡❛s✱ ❝♦♥tr✐❜✉✐♥❞♦ ♣❛r❛ q✉❡ ❡✉ r❡❛❧✐③❛ss❡ ✉♠ ❡♥♦r♠❡ s❛❧t♦ ❡✈♦❧✉t✐✈♦ ❞❡s❞❡ ❏❛♥❡✐r♦ ❞❡ 2004 ❛té ❛♦s ❞✐❛s ❞❡ ❤♦❥❡✳ ●♦st❛r✐❛ ❛✐♥❞❛

❞❡ ❞❡st❛❝❛r ❞♦✐s ❛s♣❡❝t♦s ♥❛ ♣❡rs♦♥❛❧✐❞❛❞❡ ❞♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❏♦ã♦ ❚✐❛❣♦ ▼❡①✐❛ q✉❡✱ q✉❛♥t♦ ❛ ♠✐♠✱ ❢♦r❛♠ ❞❡t❡r♠✐♥❛♥t❡s ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ ♣r♦❥❡❝t♦✿ ♣♦r ✉♠ ❧❛❞♦✱ ✉♠❛ ♥♦t✁❛✈❡❧ ❡ ✐♥❡s❣♦t✁❛✈❡❧ ❝❛♣❛❝✐❞❛❞❡ ♣❛r❛ ♠♦t✐✈❛r ❡ ✐♥❝❡♥t✐✈❛r ❡ ♣♦r ♦✉tr♦✱ s❡♠ ♥✉♥❝❛ ❞❡✐①❛r ❞❡ ♠❡ ❛❝♦♠♣❛♥❤❛r✱ ❞❡♣♦s✐t♦✉ ❡♠ ♠✐♠ t♦❞❛ ❛ ❝♦♥✜❛♥ç❛ ❛♦ ❞❛r✲♠❡ t♦t❛❧ ❧✐❜❡r❞❛❞❡ ♥❛ ❡s❝♦❧❤❛ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ tr❛❜❛❧❤♦✳ ❊st❡s ❢❛❝t♦r❡s ❢♦r❛♠✱ s❡♠ ❞✁✉✈✐❞❛ ♥❡♥❤✉♠❛✱ ♦s ✐♥❣r❡❞✐❡♥t❡s ♥❡❝❡ss✁❛r✐♦s ♣❛r❛ q✉❡ ❡st❡ ♠❛♥✉s❝r✐t♦ s❡ t♦r♥❛ss❡ ✉♠❛ r❡❛❧✐❞❛❞❡ ❡ q✉❡ ✐♠♣❧✐❝❛r❛♠ ✉♠❛ r❡s♣♦♥s❛❜✐❧✐③❛çã♦ ♣❡ss♦❛❧ ♥♦ s❡♥t✐❞♦ ❞❡ ❝♦rr❡s♣♦♥❞❡r ❛♦ ❝ré❞✐t♦ ❡♠ ♠✐♠ ❞❡♣♦s✐t❛❞♦✳

◗✉❡r♦ ❡①♣r❡ss❛r ♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛♦ Pr♦❢❡ss♦r ❉♦✉t♦r P❡❞r♦ ❈♦rt❡ ❘❡❛❧ q✉❡✱ ♥♦s ♣r✐♠❡✐r♦s t❡♠♣♦s✱ t❡✈❡ s❡♠♣r❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❡ ♣❛❝✐ê♥❝✐❛ ♣❛r❛ ❛s q✉❡stõ❡s q✉❡ ❝♦♥st❛♥t❡♠❡♥t❡ ❧❤❡ ❝♦❧♦q✉❡✐✱ ❛❧❣✉♠❛s ❞❡❧❛s t✁✙♣✐❝❛s ❞❡ q✉❡♠ ❞✁❛ ♦s ♣r✐♠❡✐r♦s ♣❛ss♦s ♥✉♠❛ ♥♦✈❛ ✁❛r❡❛ ❞❡ ❡st✉❞♦✳ ❊st♦✉ t❛♠❜é♠ ❡t❡r♥❛♠❡♥t❡ ❣r❛t♦ ❛♦ Pr♦❢❡ss♦r ❉♦✉t♦r ▼❛♥✉❡❧ ▲❡♦t❡ ❊sq✉✁✙✈❡❧ q✉❡ ♠❡ ❛❝♦♠♣❛♥❤♦✉ ❡ ❛♠♣❛r♦✉ ❛♦ ❧♦♥❣♦ ❞❡st❡ ♣r♦❥❡❝t♦✱ ♠✉✐t♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥♦ ✐♥✁✙❝✐♦❀ ♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ♣r♦❢✉♥❞♦ ❞❡ ❡①❝❡❧❡♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣r✁❛✜❝❛s ✈❡✐♦ ❛ r❡✈❡❧❛r✲s❡ ✈✐t❛❧ ♣❛r❛ ❛ ❝♦♥❝r❡t✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❯♠❛ ♣❛❧❛✈r❛ ❞❡ ❡♥♦r♠❡ ❛♣r❡ç♦ ❛✐♥❞❛ ♣❛r❛ ♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❍✐❧♠❛r ❉r②❣❛s✱ ❛ Pr♦❢❡ss♦r❛ ❉♦✉t♦r❛ ❈❧❛✉❞✐❛ ❑❧☎✉♣♣❡❧❜❡r❣ ❡ ♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❆❧❛st❛✐r ❍❛❧❧ q✉❡✱ s❡♠ ❤❡s✐t❛çã♦✱ ♠❡ ❡♥✈✐❛r❛♠ ♣♦r ❝♦rr❡✐♦ ❞❡ s✉♣❡r❢✁✙❝✐❡ ❛❧❣✉♥s ❞♦s s❡✉s ❛rt✐❣♦s✱ ❢✉♥❞❛♠❡♥t❛✐s ♥❛ ❡❧❛❜♦r❛çã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳

❆ ❝♦♥❝r❡t✐③❛çã♦ ❞❡ ✉♠❛ t❡s❡ ❞❡ ❞♦✉t♦r❛♠❡♥t♦ é s❡♠♣r❡ ♦ ❝✉❧♠✐♥❛r ❞❡ ♠✉✐t♦s ❛♥♦s ❞❡ ❛♣r❡♥✲ ❞✐③❛❣❡♠✱ r❛③ã♦ ♣❡❧❛ q✉❛❧ s❡r✐❛ ✐♥❥✉st♦ ♥ã♦ ❞❡st❛❝❛r t♦❞♦s ❛q✉❡❧❡s q✉❡✱ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❝❛♠✐♥❤❛❞❛✱ ❝♦♥tr✐❜✉✁✙r❛♠ ❞✐r❡❝t❛ ♦✉ ✐♥❞✐r❡❝t❛♠❡♥t❡ ♣❛r❛ ♦s ♠❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡♠ ♠❛t❡♠✁❛t✐❝❛✳

❙✉♠✁❛r✐♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ❡♠ ♠♦❞❡❧♦s ❧✐♥❡❛r❡s s❡r✁❛ ❛♥❛❧✐s❛❞❛ ✐♠♣♦♥❞♦ ❝♦♥❞✐çõ❡s s♦❜r❡ ❛ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦ ❡ ❛ss✉♠✐♥❞♦ ♦s ♠❛✐s ❞✐✈❡rs♦s ❝❡♥✁❛r✐♦s ♣❛r❛ ♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s✳ ❉❡ ✐♥✁✙❝✐♦✱ s❡rã♦ ❡st❛❜❡❧❡❝✐❞❛s ❝♦♥❞✐çõ❡s ❣❡r❛✐s ❞❡ s✉✜❝✐ê♥❝✐❛ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ♦s ❡rr♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉✁✙❞♦s ❝♦♠ ✈❛r✐✂❛♥❝✐❛ ✐♥✜♥✐t❛✳ ❊♠ s❡❣✉✐❞❛ ❡ ✉t✐❧✐③❛♥❞♦ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ q✉❛s❡ ❝❡rt❛ ♣❛r❛ s♦♠❛s ❞❡ ✈❛r✐✁❛✈❡✐s ❛❧❡❛t✁♦r✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡st✁❛✈❡✐s s❡r✁❛ ❡①♣❧♦r❛❞❛ ❛ s✐t✉❛çã♦ ♣❛rt✐❝✉❧❛r ❞❡ ❡rr♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ❞✐str✐❜✉✐çã♦ ❡st✁❛✈❡❧✳ ❊st❛ q✉❡stã♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ❡rr♦s s❡r✁❛ ❛✐♥❞❛ ❡st✉❞❛❞❛ ❝♦♠ ❣r❛♥❞❡ ❞❡t❛❧❤❡ ♥♦ ❝❛♣✁✙t✉❧♦ s❡❣✉✐♥t❡ r❡❝♦rr❡♥❞♦✱ ❞❡st❛ ❢❡✐t❛✱ ❛ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❡st✐♠❛t✐✈❛ s♦❜r❡ ❛ ♥♦r♠❛ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ❡ ♦ ✈❡❝t♦r ♣❛r✂❛♠❡tr♦❀ ❞❛r❡♠♦s ❡s♣❡❝✐❛❧ ê♥❢❛s❡ ❛♦ ❝❛s♦ ❡♠ q✉❡ ♦s ❡rr♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉✁✙❞♦s ❝♦♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♦♥❞❡✱ ❛ ❝✉st❛ ❞❡st❛ té❝♥✐❝❛ ✐♥♦✈❛❞♦r❛✱ s❡ ❝♦♥✲ s❡❣✉❡♠ ❡st❛❜❡❧❡❝❡r r❡s✉❧t❛❞♦s ❛♥✁❛❧♦❣♦s ❛♦s ♦❜t✐❞♦s ♣♦r ▲❛✐✱ ❘♦❜❜✐♥s ✫ ❲❡✐ ♥♦s ✜♥❛✐s ❞❛ ❞é❝❛❞❛ ❞❡ s❡t❡♥t❛✳ ❆ ❡st✐♠❛t✐✈❛ r❡❢❡r✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ s❡r✈✐r✁❛ t❛♠❜é♠ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✲ s✐stê♥❝✐❛ ❢♦rt❡ ♣❛r❛ ♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ♦♥❞❡ ♥ã♦ s❡ ✐♠♣õ❡♠ q✉❛✐sq✉❡r ❝♦♥❞✐çõ❡s ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ♦✉ ✐❞ê♥t✐❝❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ ♦s ❡rr♦s✱ ♠❛s s❡ ❛❞♠✐t❡ ❝♦♠♦ ❤✐♣✁♦t❡s❡ ❛ s✐♠❡tr✐❛ r❛❞✐❛❧ ❞♦ ✈❡❝t♦r ❞♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s✳ ❯♠❛ ✁✉❧t✐♠❛ ❛❜♦r❞❛❣❡♠ ❛ ❡st❛ t❡♠✁❛t✐❝❛ s❡r✁❛ r❡❛❧✐③❛❞❛✱ ❛ss✉♠✐♥❞♦ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❞♦ q✉❛❞r❛❞♦ ❡rr♦s ❛❧❡❛t✁♦r✐♦s s❡ ❡♥❝♦♥tr❛ ♥✉♠❛ ❞❛s três ❣r❛♥❞❡s ❝❧❛ss❡s✿ ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❋ré❝❤❡t✱ ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❲❡✐❜✉❧❧ ❡ ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ●✉♠❜❡❧✳ ❆ ❣r❛♥❞❡ ♥♦✈✐❞❛❞❡ ❛q✉✐ r❡s✐❞❡ ♥♦ ❢❛❝t♦ ❞❡ ✉s❛r♠♦s ❡①♣r❡ss❛♠❡♥t❡✱ ♥❛ ♠❛✐♦r✐❛ ❞♦s r❡s✉❧t❛❞♦s✱ ♦s ❝♦♠♣♦rt❛♠❡♥t♦s ❛ss✐♠♣t✁♦t✐❝♦s ❞❡ ❝❛✉❞❛ ❝❛r❛❝t❡r✁✙st✐❝♦s ❞❛s ❞✐str✐❜✉✐çõ❡s q✉❡ ♣❡rt❡♥❝❡♠ ❛ ❝❛❞❛ ✉♠ ❞♦s três ❞♦♠✁✙♥✐♦s ❝✐t❛❞♦s ♣r❡✈✐❛♠❡♥t❡✳

❙✉♠♠❛r②

■♥ t❤✐s ✇♦r❦✱ t❤❡ str♦♥❣ ❝♦♥s✐st❡♥❝② ♦❢ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦rs ✐♥ ♠✉❧t✐♣❧❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✇✐❧❧ ❜❡ st✉❞✐❡❞✱ ❣✐✈❡♥ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❞❡s✐❣♥ ♠❛tr✐① ❛♥❞ ✉♥❞❡r s❡✈❡r❛❧ ❛ss✉♠♣t✐♦♥s ❢♦r t❤❡ r❛♥❞♦♠ ❡rr♦rs✳ ❙t❛rt✐♥❣ ✇✐t❤ ❡st❛❜❧✐s❤✐♥❣ ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥s ♦❢ s✉✣❝✐❡♥❝② ❢♦r t❤❡ str♦♥❣ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦rs✱ ✇❤❡r❡ ✐t ✐s ❛❞♠✐tt❡❞ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛♥❞ ✐❞❡♥t✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ❢♦r r❛♥❞♦♠ ❡rr♦rs ✇✐t❤ ✐♥✜♥✐t❡ ✈❛r✐❛♥❝❡✳ ❯s✐♥❣ ❛ r❡s✉❧t ♦♥ ❛❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t st❛❜❧❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r st❛❜❧❡ ❡rr♦rs✱ t❤❡ s✐t✉❛t✐♦♥ ♦❢ st❛❜❧❡ ✐♥❞❡♣❡♥❞❡♥t ❡rr♦rs ✇✐❧❧ ❜❡ ❡①♣❧♦r❡❞✳ ❚❤✐s s✐t✉❛t✐♦♥ ♦❢ st❛❜❧❡ ❡rr♦rs ✇✐❧❧ ❜❡ st✉❞✐❡❞ ✐♥ ❞❡t❛✐❧ ♦♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❚❤❡r❡✱ ✉s✐♥❣ ❛♥ ✐♠♣♦rt❛♥t ❡st✐♠❛t❡ ❢♦r t❤❡ ♥♦r♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦r ❛♥❞ ♣❛r❛♠❡t❡r ✈❡❝t♦r✱ ❛ s♣❡❝✐❛❧ ❡♠♣❤❛s✐s ✇✐❧❧ ❜❡ ❣✐✈❡♥ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ r❛♥❞♦♠ ❡rr♦rs ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ✇✐t❤ ❣❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤r♦✉❣❤ t❤✐s ♥❡✇ t❡❝❤♥✐q✉❡✱ r❡s✉❧ts s✐♠✐❧❛r t♦ t❤♦s❡ ♦❜t❛✐♥❡❞ ❜② ▲❛✐✱ ❘♦❜❜✐♥s ✫ ❲❡✐ ✐♥ t❤❡ s❡✈❡♥t✐❡s ✇✐❧❧ ❜❡ ❡st❛❜❧✐s❤❡❞✳ ❚❤❡ ❡st✐♠❛t❡ q✉♦t❡❞ ♣r❡✈✐♦✉s❧② ✇✐❧❧ ❜❡ ✈❡r② ✉s❡❢✉❧ ✐♥ ❞❡r✐✈✐♥❣ str♦♥❣ ❝♦♥s✐st❡♥❝② r❡s✉❧ts ❢♦r ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦rs ✇❤❡♥ ❛ss✉♠♣t✐♦♥s ♦❢ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛♥❞ ✐❞❡♥t✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ r❛♥❞♦♠ ❡rr♦rs ❛r❡ ❝♦♠♣❧❡t❡❧② ❡①❝❧✉❞❡❞ ❛♥❞ ❛ r❛❞✐❛❧ s②♠♠❡tr② ❝♦♥❞✐t✐♦♥ ♦♥ r❛♥❞♦♠ ✈❡❝t♦r ✇✐❧❧ ❜❡ ❛❞♠✐tt❡❞✳ ❆ ❧❛st ❛♣♣r♦❛❝❤ t♦ t❤✐s s✉❜❥❡❝t ✇✐❧❧ ❜❡ ❝❛rr✐❡❞ ♦✉t ❛ss✉♠✐♥❣ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ r❛♥❞♦♠ ❡rr♦rs ✐s ✐♥ ♦♥❡ ♦❢ t❤❡ t❤r❡❡ ❝❧❛ss❡s✿ ♠❛①✐♠❛❧ ❞♦♠❛✐♥ ♦❢ ❛ttr❛❝t✐♦♥ ♦❢ t❤❡ ❋ré❝❤❡t ❞✐str✐❜✉t✐♦♥✱ ♦❢ t❤❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ♦❢ t❤❡ ●✉♠❜❡❧ ❞✐str✐❜✉t✐♦♥✳ ❚❤✐s ♥❡✇ ❛♣♣r♦❛❝❤ ✉s❡s t❤❡ s♣❡❝✐✜❝ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ t❛✐❧ ♦❢ ❞✐str✐❜✉t✐♦♥s ❢♦r ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❞♦♠❛✐♥s q✉♦t❡❞✳

▲✐st❛ ❞❡ s✁✙♠❜♦❧♦s ❡ ❛❜r❡✈✐❛t✉r❛s

◆ ❝♦♥❥✉♥t♦ ❞♦s ♥✁✉♠❡r♦s ♥❛t✉r❛✐s

❘✱ ❘n _{❝♦♥❥✉♥t♦ ❞♦s ♥✁✉♠❡r♦s r❡❛✐s✱ ❡s♣❛ç♦ ❡✉❝❧✐❞❡❛♥♦ n✲❞✐♠❡♥s✐♦♥❛❧} ❙n−1 _{❡s❢❡r❛ ✉♥✐t✁❛r✐❛ ❞❡ ❘}n

V

B

F✱ G✱ H σ✲✁❛❧❣❡❜r❛s

B(❘n) σ✲✁❛❧❣❡❜r❛ ❞♦s ❜♦r❡❧✐❛♥♦s ❞❡ ❘n

π(❘n_{) π✲s✐st❡♠❛ ❞❡ t♦❞♦s ♦s r❡❝t✂❛♥❣✉❧♦s ❞❡ ❘}n ▲❡❜(·) ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ s♦❜r❡ ❘

P(_·) ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

q✳❝✳ q✉❛s❡ ❝❡rt❛♠❡♥t❡

(Ω,_F) ❡s♣❛ç♦ ❛♠♦str❛❧

(Ω,_F,P) ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

✈✳❛✳ ✭✈✳❛✳✬s✮ ✈❛r✐✁❛✈❡❧ ❛❧❡❛t✁♦r✐❛ ✭✈❛r✐✁❛✈❡✐s ❛❧❡❛t✁♦r✐❛s✮

X,Y,Z, . . . ✈❛r✐✁❛✈❡✐s ❛❧❡❛t✁♦r✐❛s

e,e1,e2, . . . ❡rr♦s ❛❧❡❛t✁♦r✐♦s

Mn max(X1, . . . ,Xn)

X+_,X− _{♣❛rt❡ ♣♦s✐t✐✈❛ ❞❡X✱ ♣❛rt❡ ♥❡❣❛t✐✈❛ ❞❡ X} X= Y❞ ✱ X∼Y X ❡Y ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉✁✙❞❛s

Z_∼S_{α(σ, λ, µ) Z}t❡♠ ❞✐str✐❜✉✐çã♦ ❡st✁❛✈❡❧

X_∼N(µ, σ2_{) X t❡♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❝♦♠ ♠é❞✐❛µ❡ ✈❛r✐✂❛♥❝✐❛ σ}2

X∼χ2(ν) X t❡♠ ❞✐str✐❜✉✐çã♦χ2 ❝♦♠ν ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡

X_∼❉❊■(α, β, τ) X t❡♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦ ✐♥✜♥✐t♦ (X1, . . . ,Xn) ✈❡❝t♦r ❛❧❡❛t✁♦r✐♦

✐✳✐✳❞✳ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉✁✙❞❛s

❊(X) ✈❛❧♦r ♠é❞✐♦ ❞❡X

❊|X_|p ♠♦♠❡♥t♦ ❛❜s♦❧✉t♦ ❞❡ ♦r❞❡♠p ❞❡X

❱(X) ✈❛r✐✂❛♥❝✐❛ ❞❡X

L_p(Ω,F,P₎ ❡s♣❛ç♦L_p

s✉♣ q✳❝✳ s✉♣r❡♠♦ q✉❛s❡ ❝❡rt♦

① ▲✐st❛ ❞❡ s✁✙♠❜♦❧♦s ❡ ❛❜r❡✈✐❛t✉r❛s

IA ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r ❞♦ ❝♦♥❥✉♥t♦ A

ϕX ❢✉♥çã♦ ❝❛r❛❝t❡r✁✙st✐❝❛ ❞❛ ✈❛r✐✁❛✈❡❧ ❛❧❡❛t✁♦r✐❛ X

❢✳❞✳ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦

❙✉♣♣(F) s✉♣♦rt❡ ❞❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ F FX✱ FX ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ X✱ ❝❛✉❞❛ ❞❡FX fX ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ X

FX1,...,Xn ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❝♦♥❥✉♥t❛ ❞❡ (X1, . . . ,Xn)

F←− ❢✉♥çã♦ q✉❛♥t✐❧

❆(x) ❢✉♥çã♦ ❛✉①✐❧✐❛r

Φα(x)✱ Λ(x)✱ Ψα(x) ❞✐str✐❜✉✐çõ❡s ❞❡ ✈❛❧♦r❡s ❡①tr❡♠♦s

D_(Φα)✱ D_(Λ)✱ D_(Ψα) ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❡ _Φα✱ _Λα✱_Ψα

−→ ❝♦♥✈❡r❣ê♥❝✐❛

q✳❝✳

−→ ❝♦♥✈❡r❣ê♥❝✐❛ q✉❛s❡ ❝❡rt❛

P

−→ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡

L_p

−→ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ♠♦♠❡♥t♦ ❞❡ ♦r❞❡♠ p

❞

−→ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❞✐str✐❜✉✐çã♦ ♦✉ ❡♠ ❧❡✐

M_n×κ(❘₎ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s r❡❝t❛♥❣✉❧❛r❡s ❞❡ _n×κ s♦❜r❡ ❘ I✱ O ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱ ♠❛tr✐③ ♥✉❧❛

AT✱ A−1 ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ A✱ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ A

h ·, _{· i} ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❛♥✁♦♥✐❝♦

x_⊥y x ♦rt♦❣♦♥❛❧ ❛y

P_❋x ♣r♦❥❡❝çã♦ ♦rt♦❣♦♥❛❧ ❞♦ ✈❡❝t♦r xs♦❜r❡ ❋

❝❛r(A) ❝❛r❛❝t❡r✁✙st✐❝❛ ❞❛ ♠❛tr✐③A

◆✉❝(A)✱ ■♠(A) ♥✁✉❝❧❡♦ ❞❡A✱ ❡s♣❛ç♦ ✐♠❛❣❡♠ ❞❡A

dim (❋) ❞✐♠❡♥sã♦ ❞♦ s✉❜❡s♣❛ç♦ ✈❡❝t♦r✐❛❧ ❋

❙♣❡❝(A) ❡s♣❡❝tr♦ ❞❛ ♠❛tr✐③ A

ρ(A) r❛✐♦ ❡s♣❡❝tr❛❧ ❞❛ ♠❛tr✐③ A

||x_|| ♥♦r♠❛ ❡✉❝❧✐❞❡❛♥❛ ❞♦ ✈❡❝t♦r x

||x_||∞ ♥♦r♠❛ ❞♦ ♠✁❛①✐♠♦ ❞♦ ✈❡❝t♦r x

|||A_||| sup_x6=0 ||

Ax|| ||x||

|x_|min min _|x1|, . . . ,|xκ|

➔n ♥✁✉♠❡r♦s ❞❡ ❇❡r♥♦✉❧❧✐

lim sup✱ lim inf✱ lim ❧✐♠✐t❡ s✉♣❡r✐♦r✱ ❧✐♠✐t❡ ✐♥❢❡r✐♦r✱ ❧✐♠✐t❡

f =O(g)✱ x→x0 f ❢r❛❝❛♠❡♥t❡ ✐♥❢❡r✐♦r ❛g❡♠ x0 f _≍g✱x_→x0 f ❞♦ ♠❡s♠♦ ❣r❛✉ ❞❡g❡♠ x0 f =o(g)✱ x→x0 f ❢♦rt❡♠❡♥t❡ ✐♥❢❡r✐♦r ❛g ❡♠x0

f _∼g✱x_→x0 f ✭❛ss✐♠♣t♦t✐❝❛♠❡♥t❡✮ ❡q✉✐✈❛❧❡♥t❡ ❛g ❡♠x0

Γ(x) ❢✉♥çã♦ ❣❛♠❛

▲✐st❛ ❞❡ s✁✙♠❜♦❧♦s ❡ ❛❜r❡✈✐❛t✉r❛s ①✐

❡r❢(x) ❢✉♥çã♦ ❡rr♦

⑨n(x) ❢✉♥çã♦ ❞❡ ❇❡ss❡❧ ✭❞❡ 1❛ ❡s♣é❝✐❡✮

❼n(x) ❢✉♥çã♦ ❞❡ ❇❡ss❡❧ ♠♦❞✐✜❝❛❞❛

➚k ✐♥t❡❣r❛❧ ❞❡ ❊✉❧❡r✲▼❛s❝❤❡r♦♥✐

log_mx log₁x= log (x)✱ log₂x= log (log (x)), . . .

exp_mx exp₁x= exp (x)✱ exp₂x= exp (exp (x)), . . .

[x] ♠❛✐♦r ✐♥t❡✐r♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛x

δip s✁✙♠❜♦❧♦ ❞❡ ❑r♦♥❡❝❦❡r

(α)n s✁✙♠❜♦❧♦ ❞❡ P♦❝❤❤❛♠♠❡r

p❸q(a1, . . . , ap;b1, . . . , bq;x) ❢✉♥çã♦ ❤✐♣❡r❣❡♦♠étr✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛

p❸qe (a1, . . . , ap;b1, . . . , bq;x) ❢✉♥çã♦ ❤✐♣❡r❣❡♦♠étr✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ r❡❣✉❧❛r✐③❛❞❛

D ❛❞❡rê♥❝✐❛ ❞❡ D

max

16j6naj max(a1, . . . , an)

max

❍ f ♠✁❛①✐♠♦ ❞❛ ❢✉♥çã♦ f ♥♦ ❝♦♥❥✉♥t♦ ❍

❣r❛❞(f) ❣r❛❞✐❡♥t❡ ❞❡ f

s✐❣♥(t) ❢✉♥çã♦ s✐♥❛❧

xinf✱ xsup ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❞♦ s✉♣♦rt❡✱ ❡①tr❡♠♦ ❞✐r❡✐t♦ ❞♦ s✉♣♦rt❡

R_0,R_α,R_−∞ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ✈❛r✐❛çã♦ ❧❡♥t❛✱ r❡❣✉❧❛r✱ r✁❛♣✐❞❛ ♥♦ ✐♥✜♥✐t♦

ℓ# ❝♦♥❥✉❣❛❞❛ ❞❡ ❇r✉✐❥♥ ❞❛ ❢✉♥çã♦ ❞❡ ✈❛r✐❛çã♦ ❧❡♥t❛ℓ

❊▼◗ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s

Y_n=X_nβ+e_n ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r

Xn ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦

β ✈❡❝t♦r ♣❛r✂❛♠❡tr♦

e

β ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s

e_n ✈❡❝t♦r ❞♦s ❡rr♦s

✁■♥❞✐❝❡

▲✐st❛ ❞❡ s✁✙♠❜♦❧♦s ❡ ❛❜r❡✈✐❛t✉r❛s ✐①

Pr❡❢✁❛❝✐♦ ①✈

✶ Pr❡❧✐♠✐♥❛r❡s ✶

✶✳✶ ❚❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❊❧❡♠❡♥t♦s ❞❡ ❛♥✁❛❧✐s❡ ♠❛tr✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✸ ❚✁♦♣✐❝♦s ❞❡ ❛♥✁❛❧✐s❡ r❡❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✹ ❉✐str✐❜✉✐çõ❡s ❡st✁❛✈❡✐s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✶✳✺ ❉♦♠✁✙♥✐♦s ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✶✳✻ ❆♥✁❛❧✐s❡ ❞❡ r❡❣r❡ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✷ ❆ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ✼✶

✷✳✶ ❈♦♥❞✐çõ❡s ❣❡r❛✐s ❞❡ s✉✜❝✐ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✷✳✷ ❈♦♥❞✐çõ❡s ❞❡ s✉✜❝✐ê♥❝✐❛ ♣❛r❛ ❡rr♦s ❡st✁❛✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾

✸ ◆❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ❡rr♦s ✽✸

✸✳✶ ❖ ❝❛s♦ ♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✷ ❖ ❝❛s♦ 0< α <2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

✸✳✸ ❙♦❜ ♣❛r✂❛♠❡tr♦s σi✱ λi ❡ µi ✈❛r✐✁❛✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

✹ ❯♠❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ✶✵✹

✹✳✶ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ Zn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✹✳✷ ❆ ❝♦♥s✐stê♥❝✐❛ ❞♦ ❡st✐♠❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✹✳✸ ❆ ❡①t❡♥sã♦ ❛♦ ❝❛s♦ α= 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✻

✺ ❊♠ ❞♦♠✁✙♥✐♦s ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛✐s ✶✷✷

①✐✈ ✁■◆❉■❈❊

✻ ❊st✐♠❛❞♦r❡s ♥ã♦✲❧✐♥❡❛r❡s ✶✸✸

✻✳✶ ❱❛r✐✁❛✈❡✐s ❛❧❡❛t✁♦r✐❛s ❝♦♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹ ✻✳✷ ❆ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺ ✻✳✸ ◆❛ s✐♠❡tr✐❛ r❛❞✐❛❧ ❞❡ ❡rr♦s ✐♥❞❡♣❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✾ ✻✳✹ ❊①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸

❆♣ê♥❞✐❝❡ ✶✹✻

❇✐❜❧✐♦❣r❛✜❛ ✶✻✵

✁■♥❞✐❝❡ ❞❡ ❆✉t♦r❡s ✶✻✻

Pr❡❢✁❛❝✐♦

❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡❝t✐✈♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ é ❡st❛❜❡❧❡❝❡r ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ❡♠ ♠♦❞❡❧♦s ❧✐♥❡❛r❡s ❛ss✉♠✐♥❞♦ ❛s ♠❛✐s ❞✐✈❡rs❛s ❤✐♣✁♦t❡s❡s ♣❛r❛ ♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s ❡ ✐♠♣♦♥❞♦ ❝♦♥❞✐çõ❡s s♦❜r❡ ❛ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣✁✙t✉❧♦✱ s❡rã♦ r❡❝♦r❞❛❞❛s ❛❧❣✉♠❛s ♥♦çõ❡s ❡❧❡♠❡♥t❛r❡s ❞❡ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ❞✐str✐❜✉✐çõ❡s ❡st✁❛✈❡✐s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s ❡ t❡♦r✐❛ ❞❡ ✈❛❧♦r❡s ❡①tr❡♠♦s ❡①♣♦♥❞♦✲s❡✱ ♣♦r ✉♠ ❧❛❞♦✱ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❡✜♥✐çõ❡s q✉❡ s❡r✈✐rã♦ ❞❡ ❜❛s❡ ♣❛r❛ ♦s ✈✁❛r✐♦s ❝❛♣✁✙t✉❧♦s ♣♦st❡r✐♦r❡s ❡✱ ♣♦r ♦✉tr♦✱ ✉♠❛ ❝♦❧❡❝t✂❛♥❡❛ ❞❡ ✁✉t❡✐s r❡s✉❧t❛❞♦s ❝♦♠ ✐♠♣♦rt✂❛♥❝✐❛ ❞❡❝✐s✐✈❛ ♥♦ ❝♦♥t❡✁✉❞♦ ♦r✐❣✐♥❛❧✳ ❉❡ ❞❡st❛❝❛r q✉❡ ♥❛ s❡❣✉♥❞❛ ❡ t❡r❝❡✐r❛ s❡❝çõ❡s ❞♦ ♣r✐♠❡✐r♦ ❝❛♣✁✙t✉❧♦ s❡rã♦ r❡❧❡♠❜r❛❞♦s t❛♠❜é♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❛♥✁❛❧✐s❡ ♠❛tr✐❝✐❛❧ ❡ r❡❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ ❢✉♥❝✐♦♥❛r❛♠ ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ❛✉①✐❧✐❛r❡s ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ t❡①t♦✳ ❖ ♣r✐♠❡✐r♦ ❝❛♣✁✙t✉❧♦ é ✜♥❛❧✐③❛❞♦ ❝♦♠ ✉♠❛ ✁✉❧t✐♠❛ s❡❝çã♦ ❞❡❞✐❝❛❞❛ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❛ ❛♥✁❛❧✐s❡ ❞❡ r❡❣r❡ssã♦ ❞❛♥❞♦ ❡s♣❡❝✐❛❧ ❞❡st❛q✉❡ ❛ ♠♦t✐✈❛çã♦ q✉❡ ❝♦♥❞✉③ ❛♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r✳

◆♦ s❡❣✉♥❞♦ ❝❛♣✁✙t✉❧♦✱ s❡rã♦ ❛♥✉♥❝✐❛❞♦s ♦s ♣r✐♠❡✐r♦s r❡s✉❧t❛❞♦s ♦r✐❣✐♥❛✐s q✉❡ ❝♦♥st✐t✉❡♠ ✉♠❛ ❡①t❡♥sã♦ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ s✉✜❝✐ê♥❝✐❛ ♣❛r❛ ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s✱ ❡st❛❜❡❧❡❝✐❞❛s ♥❛ ❞é❝❛❞❛ ❞❡ ♥♦✈❡♥t❛ ♣♦r ▼✐♥❣③❤♦♥❣ ❏✐♥ ❡ ❳✐r✉ ❈❤❡♥✱ ❛♦ ❝❛s♦ ❡♠ q✉❡ ♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s ❞♦ ♠♦❞❡❧♦ ❧✐♥❡❛r ♥ã♦ ♣♦ss✉❡♠ ♠♦♠❡♥t♦s ❛❜s♦❧✉t♦s✳ ❆✐♥❞❛ ♥❡st❡ ❝❛♣✁✙t✉❧♦ ❡ s❡❣✉✐♥❞♦ ✉♠ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♣❛r❛ ♦ ❳■❱ ❈♦♥❣r❡ss♦ ❆♥✉❛❧ ❞❛ ❙P❊ s❡r✁❛ ❡❧❛❜♦r❛❞❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛❧t❡r♥❛t✐✈❛ ❛ ❛♥t❡r✐♦r✱ ❛ss✉♠✐♥❞♦ ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s ❞♦ ♠♦❞❡❧♦ ❧✐♥❡❛r✳

◆♦ t❡r❝❡✐r♦ ❝❛♣✁✙t✉❧♦ ❡ ❛✐♥❞❛ ❞❡♥tr♦ ❞❛ t❡♠✁❛t✐❝❛ ❞❛ ❡st❛❜✐❧✐❞❛❞❡✱ s❡r✁❛ ❡st✉❞❛ ❝♦♠ ❣r❛♥❞❡ ❞❡t❛❧❤❡ ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❡ ❛ ❝♦♥s✐stê♥❝✐❛ ❡♠ ♠♦♠❡♥t♦ ❞❡ ♦r❞❡♠ 0 < r 6 2 ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s

q✉❛❞r❛❞♦s ♣♦r ✉♠ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛t✐✈❛✳ ❆ ❡st✐♠❛t✐✈❛ ❡♠ ❝❛✉s❛ s❡r✁❛ ♦❜t✐❞❛ s♦❜r❡ ❛ ♥♦r♠❛ ❞❛ ❞✐❢❡✲ r❡♥ç❛ ❡♥tr❡ ♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ❡ ♦ ✈❡❝t♦r ♣❛r✂❛♠❡tr♦ r❡❝♦rr❡♥❞♦ ❛♣❡♥❛s ❛ ♦♣❡r❛çõ❡s ❞❡ ❝✁❛❧❝✉❧♦ ♠❛tr✐❝✐❛❧✳ ❆tr❛✈és ❞❡st❛ ♣♦❞❡r♦s❛ té❝♥✐❝❛✱ ❝♦♥s❡❣✉❡♠✲s❡ ❡st❛❜❡❧❡❝❡r r❡s✉❧t❛❞♦s ❛♥✁❛❧♦❣♦s ❛♦s ♦❜t✐❞♦s ♣♦r ▲❛✐✱ ❘♦❜❜✐♥s ✫ ❲❡✐ ♥♦s ✜♥❛✐s ❞❛ ❞é❝❛❞❛ ❞❡ s❡t❡♥t❛ ♥✉♠ ❝❡♥✁❛r✐♦ ❡♠ q✉❡ ♦s ❡rr♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉✁✙❞♦s ❝♦♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ❯s✉❢r✉✐♥❞♦ ❛✐♥❞❛ ❞❛s ♠❛❣♥✁✙✜❝❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❞✐str✐❜✉✐çõ❡s ❡st✁❛✈❡✐s✱ ❝♦♥s❡❣✉❡♠✲s❡ ❣❡♥❡r❛❧✐③❛r ♦s r❡s✉❧t❛❞♦s ❞♦ ❝❛s♦ ♥♦r♠❛❧ ❛ s✐t✉❛✲ çã♦ ❡♠ q✉❡ ❛ ✈❛r✐✂❛♥❝✐❛ ❞♦s ❡rr♦s ♥ã♦ ❡st✁❛ ❞❡✜♥✐❞❛ ❛ss✉♠✐♥❞♦ ❛♣❡♥❛s ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡st❛s ✈❛r✐✁❛✈❡✐s ❛❧❡❛t✁♦r✐❛s✳

◆♦ q✉❛rt♦ ❝❛♣✁✙t✉❧♦✱ s❡r✁❛ ❛♥❛❧✐s❛❞❛ ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❡ ❛ ❝♦♥s✐stê♥❝✐❛ ❡♠ ♠♦♠❡♥t♦ ❞❡ ♦r❞❡♠ s > 0 ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ❡①❝❧✉✐♥❞♦ q✉❛✐sq✉❡r ❤✐♣✁♦t❡s❡s ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ♦✉

①✈✐ Pr❡❢✁❛❝✐♦

❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s ❣❡♥❡r❛❧✐③❛❞❛s ❝♦♥s❡❣✉❡✲s❡ ✉♠❛ ❡❧❡❣❛♥t❡ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ❛ ❝♦♥s✐stê♥❝✐❛ ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ♥✉♠ ❧❡q✉❡ ❞❡ ❤✐♣✁♦t❡s❡s ♣♦✉❝♦ ❤❛❜✐t✉❛❧✳ ◆♦ ✜♥❛❧✱ ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣r✁❛t✐❝♦s s❡rã♦ ❡①♣♦st♦s ❝♦♠ ❛ r❡s♣❡❝t✐✈❛ ❛♣❧✐❝❛çã♦ ❞❛ t❡♦r✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳

◆♦ q✉✐♥t♦ ❝❛♣✁✙t✉❧♦ ❡ t❡♥❞♦ ♣♦r ❜❛s❡ ✉♠ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♥♦ ❳■■■ ❈♦♥❣r❡ss♦ ❆♥✉❛❧ ❞❛ ❙P❊✱ s❡r✁❛ ❢❡✐t♦ ✉♠ ✁✉❧t✐♠♦ ❡st✉❞♦ ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❡ ❡♠ ♠é❞✐❛ q✉❛❞r✁❛t✐❝❛ ❞♦ ❡st✐♠❛❞♦r ❞♦s ♠✁✙♥✐♠♦s q✉❛❞r❛❞♦s ❛❞♠✐t✐♥❞♦ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❞♦ q✉❛❞r❛❞♦ ❞♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s s❡ ❡♥❝♦♥tr❛ ♥✉♠❛ ❞❛s três ❣r❛♥❞❡s ❝❧❛ss❡s✿ ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❋ré❝❤❡t✱ ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❲❡✐❜✉❧❧ ❡ ❞♦♠✁✙♥✐♦ ❞❡ ❛tr❛❝çã♦ ♠❛①✐♠❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ●✉♠❜❡❧✳ ❆ ❣r❛♥❞❡ ♥♦✈✐❞❛❞❡ ❛q✉✐ r❡s✐❞✐r✁❛ ♥♦ ❢❛❝t♦ ❞❡ ✉t✐❧✐③❛r♠♦s✱ ♥❛ ♠❛✐♦r✐❛ ❞♦s r❡s✉❧t❛❞♦s✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♠♣t✁♦t✐❝♦ ❡s♣❡❝✁✙✜❝♦ ❞❛s ❞✐str✐❜✉✐çõ❡s q✉❡ ♣❡rt❡♥❝❡♠ ❛ ❝❛❞❛ ✉♠ ❞♦s três ❞♦♠✁✙♥✐♦s ♣r❡✈✐❛♠❡♥t❡ ❝✐t❛❞♦s✳

❋✐♥❛❧♠❡♥t❡✱ ♥♦ ✁✉❧t✐♠♦ ❝❛♣✁✙t✉❧♦ s❡r✁❛ ❛♥❛❧✐s❛❞❛ ❛ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡ ❞❡ ❡st✐♠❛❞♦r❡s ♥ã♦✲❧✐♥❡❛r❡s ♥✉♠ ♠♦❞❡❧♦ ❝♦♥❝r❡t♦ ♣❛r❛ ❛s ♦❜s❡r✈❛çõ❡s✱ ♦♥❞❡ s❡ ❞❡s❡♥✈♦❧✈❡r✁❛ ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ❛❞♠✐t✐♥❞♦ q✉❡ ♦s ❡rr♦s ❛❧❡❛t✁♦r✐♦s s❛t✐s❢❛③❡♠ ❤✐♣✁♦t❡s❡s ❛❞❡q✉❛❞❛s✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ q✉❡ ♣♦ss✉❡♠ ❞❡❝❛✐✲ ♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦ ✐♥✜♥✐t♦✳ ❖ ❝❛s♦ ❡♠ q✉❡ ♦s ❡rr♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉✁✙❞♦s ❝♦♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ é ❡st✉❞❛❞♦ ♥✉♠❛ s❡❝çã♦ ❛ ♣❛rt❡❀ ❛ t❡r♠✐♥❛r s❡rã♦ ❡①✐❜✐❞♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣r✁❛t✐❝♦s s❡♥❞♦ ❢❡✐t❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦s r❡s♣❡❝t✐✈♦s r❡s✉❧t❛❞♦s t❡✁♦r✐❝♦s✳

❋❛r♦✱ ❋❡✈❡r❡✐r♦ ❞❡ 2007✱

❈❛♣✁✙t✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

✶✳✶ ❚❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s

❍✐st♦r✐❝❛♠❡♥t❡✱ ❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s t❡✈❡ ❛ s✉❛ ♦r✐❣❡♠ ❡♠ ❥♦❣♦s ❞❡ ❛③❛r ❡ ❛♣♦st❛s✱ q✉❡ ❣♦③❛✈❛♠ ❞❡ ❡♥♦r♠❡ ♣♦♣✉❧❛r✐❞❛❞❡ ♥♦ sé❝✉❧♦ ❳❱■■✳ ❆s ♣r✐♠❡✐r❛s t❡♥t❛t✐✈❛s ♣❛r❛ ❢♦r♠❛❧✐③❛r ❛s ✐❞❡✐❛s ✐♥t✉✐t✐✈❛s ❛ss♦❝✐❛❞❛s ❛ ❡st❡ t✐♣♦ ❞❡ ❞✐s♣✉t❛s✱ ❞❡✈❡♠✲s❡ ❛♦s ❢❛♠♦s♦s ♠❛t❡♠✁❛t✐❝♦s ❢r❛♥❝❡s❡s ❇❧❛✐s❡ P❛s✲ ❝❛❧ ❡ P✐❡rr❡ ❞❡ ❋❡r♠❛t✳ ❆♦ q✉❡ ♣❛r❡❝❡✱ ♥♦ ❧♦♥❣✁✙♥q✉♦ ❛♥♦ ❞❡ 1654✱ ♦ ♥♦❜r❡ ❢r❛♥❝ês ❆♥t♦✐♥❡ ●♦♠❜❛✉❞

✭❈❤❡✈❛❧✐❡r ❞❡ ▼éré✮ ❝❤❛♠♦✉ ❛ ❛t❡♥çã♦ ❞❡ P❛s❝❛❧ ♣❛r❛ ✉♠❛ ❛♣❛r❡♥t❡ ❝♦♥tr❛❞✐çã♦ ♥✉♠ ✧❥♦❣♦ ❞❡ ❞❛❞♦s✧✳ ❖ ❥♦❣♦ ❝♦♥s✐st✐❛ ❡♠ ❧❛♥ç❛r ✉♠ ♣❛r ❞❡ ❞❛❞♦s24 ✈❡③❡s ❡ ♦ ♣r♦❜❧❡♠❛ ❡st❛✈❛ ❡♠ ❞❡❝✐❞✐r s❡ ❡r❛ ❝♦rr❡❝t♦

❛♣♦st❛r ❛ ♠❡s♠❛ ✐♠♣♦rt✂❛♥❝✐❛ ❛ ❢❛✈♦r ♦✉ ❝♦♥tr❛ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ✧❞✉♣❧♦ s❡✐s✧✱ ♥♦s ✈✐♥t❡ ❡ q✉❛tr♦ ❧❛♥ç❛♠❡♥t♦s✳ ❊st❡ ❡ ♦✉tr♦s ♣r♦❜❧❡♠❛s ❝♦❧♦❝❛❞♦s ♣♦r ●♦♠❜❛✉❞ ♠♦t✐✈❛r❛♠ ✉♠❛ tr♦❝❛ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ P❛s❝❛❧ ❡ ❋❡r♠❛t✱ ♥❛ q✉❛❧ s❡ ❡st❛❜❡❧❡❝❡r❛♠ ♦s ♣r✐♥❝✁✙♣✐♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡st❛ t❡♦r✐❛✳ ❯♠ ♣r♦❢❡ss♦r ❞❡ ▲❡✐❜♥✐③✱ ❈❤r✐st✐❛♥ ❍✉②❣❡♥s✱ ❝♦♥❤❡❝❡❞♦r ❞♦ ❝♦♥t❡✁✉❞♦ ❞❡ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♣✉❜❧✐❝♦✉ ❡♠ 1657 ♦ ❧✐✈r♦ ❉❡ ❘❛t✐♦❝✐♥✐✐s ✐♥ ▲✉❞♦ ❆❧❡❛❡✱ ✉♠ tr❛t❛❞♦ ❝♦♠ ♣r♦❜❧❡♠❛s ❛ss♦❝✐❛❞♦s ❛♦s

❥♦❣♦s ❞❡ ❛③❛r ❡ ❞❡s❞❡ ❡♥tã♦✱ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣r♦ss❡❣✉✐✉ ❝♦♠ ♦ ❝♦♥tr✐❜✉t♦ ❞❡ ❏❛❦♦❜ ❇❡r♥♦✉❧❧✐ ❡ ❆❜r❛❤❛♠ ❉❡ ▼♦✐✈r❡✳ ❊♠ 1812✱ P✐❡rr❡ ❙✐♠♦♥ ▲❛♣❧❛❝❡ ✐♥tr♦❞✉③✐✉✱ ♥❛ s✉❛ ❚❤é♦r✐❡ ❆♥❛❧②t✐q✉❡ ❞❡s

Pr♦❜❛❜✐❧✐tés✱ ✉♠ ❣r❛♥❞❡ ♥✁✉♠❡r♦ ❞❡ ❝♦♥❝❡✐t♦s ❡ té❝♥✐❝❛s ♠❛t❡♠✁❛t✐❝❛s✱ ❞❡♠♦♥str❛♥❞♦ ❛ s✉❛ ❛♣❧✐❝❛çã♦ ❡♠ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❝✐❡♥t✁✙✜❝♦s ❡ ♣r✁❛t✐❝♦s ✭❛ t❡♦r✐❛ ❞♦s ❡rr♦s✱ ❛ ♠❛t❡♠✁❛t✐❝❛ ❛❝t✉❛r✐❛❧ ❡ ❛ ♠❡❝✂❛♥✐❝❛ ❡st❛t✁✙st✐❝❛ ❢♦r❛♠ ❛❧❣✉♠❛s ❞❛s ✁❛r❡❛s q✉❡ s❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ♥♦ sé❝✉❧♦ ❳■❳ ❛ ❝✉st❛ ❞❛s ✐❞❡✐❛s ❛♣r❡s❡♥✲ t❛❞❛s ♣♦r ▲❛♣❧❛❝❡✮✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❢♦r♠✉❧♦✉ ♦ q✉❡ s❡ ❝♦st✉♠❛ ❞❡s✐❣♥❛r ♣♦r ❞❡✜♥✐çã♦ ❝❧✁❛ss✐❝❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✿ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❛❝♦♥t❡❝✐♠❡♥t♦ é ♦ q✉♦❝✐❡♥t❡ ❞♦ ♥✁✉♠❡r♦ ❞❡ ❝❛s♦s ❢❛✈♦r✁❛✈❡✐s ❛ ❡ss❡ ❛❝♦♥t❡❝✐♠❡♥t♦ s♦❜r❡ ♦ ♥✁✉♠❡r♦ ❞❡ ❝❛s♦s ♣♦ss✁✙✈❡✐s✱ ❞❡s❞❡ q✉❡ ❡st❡s s❡❥❛♠ ❡q✉✐♣r♦✈✁❛✈❡✐s ✭♦✉ ✐❣✉❛❧✲ ♠❡♥t❡ ♣r♦✈✁❛✈❡✐s✮✳ ❈♦♥t✉❞♦✱ ❡st❛ ❞❡✜♥✐çã♦ ♣❛r❛ ❛❧é♠ ❞❡ r❡str✐t✐✈❛ é ✐♥❛❞❡q✉❛❞❛ ❥✁❛ q✉❡ ♥ã♦ ❞❡✜♥❡ r❡❛❧♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ s❡♥❞♦ ❛♣❡♥❛s ✉♠ ♠ét♦❞♦ ♣r✁❛t✐❝♦ ❞❡ ❝✁❛❧❝✉❧♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛❝♦♥t❡❝✐♠❡♥t♦s s✐♠♣❧❡s✱ ❧✐♠✐t❛❞♦ ❛♦s ❝♦♥❥✉♥t♦s ✜♥✐t♦s ❞❡ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣♦ss✁✙✈❡✐s✳ ❊❢❡❝t✐✈❛✲ ♠❡♥t❡✱ ♦ ❣r❛♥❞❡ ♣r♦❜❧❡♠❛ r❡s✐❞✐❛ ♥♦ ❢❛❝t♦ ❞❡ s❡ ♣♦❞❡r ❞❡✜♥✐r ✧❛♦ ❛❝❛s♦✧❞❛ ♠❛♥❡✐r❛ q✉❡ q✉✐séss❡♠♦s ❡ ❛s ❞✐❢❡r❡♥t❡s ❞❡✜♥✐çõ❡s ❝♦♥❞✉③✐❛♠✲♥♦s ❛ ❞✐❢❡r❡♥t❡s r❡s♣♦st❛s✳ ❊♠ 1889✱ ❏♦s❡♣❤ ❇❡rtr❛♥❞ ❡❞✐t♦✉

✷ ✶✳ Pr❡❧✐♠✐♥❛r❡s

q✉❡ ✉❧tr❛♣❛ss❛ss❡ t♦❞❛s ❡st❛s ❞✐✜❝✉❧❞❛❞❡s✳

❖ ❛ss✉♥t♦ ❢♦✐ ❞❡✜♥✐t✐✈❛♠❡♥t❡ r❡s♦❧✈✐❞♦ ❥✁❛ ♥♦ sé❝✉❧♦ ❳❳✱ ♥♦ ❛♥♦ ❞❡ 1933✱ ♣♦r ✉♠ ♥♦♠❡ ✐♥✲

❝♦♥t♦r♥✁❛✈❡❧ ♥❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✿ ❆♥❞r❡✐ ◆✐❦♦❧❛❡✈✐t❝❤ ❑♦❧♠♦❣♦r♦✈✳ ❊st❡ ♠❛t❡♠✁❛t✐❝♦ r✉ss♦ ❛①✐♦♠❛t✐③♦✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❛ s✉❛ ♠♦♥♦❣r❛✜❛ ❋♦✉♥❞❛t✐♦♥s ♦❢ t❤❡ ❚❤❡♦r② ♦❢ Pr♦❜❛❜✐❧✐t② ❧❛♥ç❛♥❞♦✱ ❛ss✐♠✱ ❜❛s❡s s✁♦❧✐❞❛s ♣❛r❛ ❛ ♠♦❞❡r♥❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✳ ❉❡s❞❡ ❡♥tã♦✱ ❛s ✐❞❡✐❛s ❞❡ ❆✳ ◆✳ ❑♦❧♠♦❣♦r♦✈ ❢♦r❛♠ s❡♥❞♦ r❡✜♥❛❞❛s✱ s❡♠♣r❡ s♦❜ ♦ ❡st✐♠✉❧♦ ❞❛s ❛♣❧✐❝❛çõ❡s ✭❛ ❡st❛t✁✙st✐❝❛ é ✉♠ ❞♦s ❡①❡♠♣❧♦s ♥♦t✁❛✈❡✐s ♣❛r❛ ❛❧é♠ ❞❡ ♦✉tr♦s tã♦ ❞✐✈❡rs✐✜❝❛❞♦s ❝♦♠♦ ❛ ❡❝♦♥♦♠✐❛✱ ♣s✐❝♦❧♦❣✐❛✱ ❣❡♥ét✐❝❛ ♦✉ ❡♥❣❡♥❤❛r✐❛✮ ❡ ❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s é✱ ❤♦❥❡ ❡♠ ❞✐❛✱ ♣❛rt❡ ❞❡ ✉♠❛ t❡♦r✐❛ ♠❛✐s ❣❡r❛❧ ❝❤❛♠❛❞❛ t❡♦r✐❛ ❞❛ ♠❡❞✐❞❛✳

❆ ♣❛❧❛✈r❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉t✐❧✐③❛çã♦ ❝♦♠✉♠ ♥♦ ✈♦❝❛❜✉❧✁❛r✐♦ q✉♦t✐❞✐❛♥♦✱ ❡st✁❛ ❛ss♦❝✐❛❞❛ ❛ s✐t✉❛çõ❡s ❡♠ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♥♦ ♠♦♠❡♥t♦✱ ♥ã♦ ♥♦s ♣❡r♠✐t❡ s❛❜❡r ❛ ❡✈♦❧✉çã♦ ❢✉t✉r❛✱ ✐st♦ é✱ s❛❜❡♠♦s q✉❡ ❤✁❛ ✈✁❛r✐♦s r❡s✉❧t❛❞♦s ♣♦ss✁✙✈❡✐s ♠❛s ❞❡s❝♦♥❤❡❝❡♠♦s q✉❛❧ ❞❡❧❡s s❡ ✐r✁❛ r❡❛❧✐③❛r✳ ❆ ❡st❡ t✐♣♦ ❞❡ ❢❡♥✁♦♠❡♥♦s ♦♥❞❡ ♦ ❡st✉❞♦ t♦r♥❛ ✐♥❞✐s♣❡♥s✁❛✈❡❧ ♦ ✉s♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❝❤❛♠❛♠♦s✱ ❡♠ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✶ ❯♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛ é ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ♦♥❞❡✿ ✭❊✶✮ ❈♦♥❤❡❝❡♠♦s t♦❞♦s ♦s s❡✉s ♣♦ss✁✙✈❡✐s r❡s✉❧t❛❞♦s✳

✭❊✷✮ ◆ã♦ s❡ ❝♦♥❤❡❝❡ ❛♥t❡❝✐♣❛❞❛♠❡♥t❡ q✉❛❧ ❞♦s r❡s✉❧t❛❞♦s ♣♦ss✁✙✈❡✐s ✈❛✐ ♦❝♦rr❡r ❝❛❞❛ ✈❡③ q✉❡ é ❡❢❡❝✲ t✉❛❞❛✳

✭❊✸✮ ❆ s✉❛ r❡♣❡t✐çã♦ s❡ ♣♦❞❡ r❡❛❧✐③❛r ❡♠ ❝♦♥❞✐çõ❡s ❛♥✁❛❧♦❣❛s✳

❊♠ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s é ❡st✉❞❛❞❛ ❛ ✐♥❝❡rt❡③❛ ❞❡ ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛✱ ♣❡❧♦ q✉❡ é ❝♦♥✈❡♥✐❡♥t❡ ❛ss♦❝✐❛r ❛ ❝❛❞❛ ❡①♣❡r✐ê♥❝✐❛✱ ✉♠ ❝♦♥❥✉♥t♦ Ω ❞❡ r❡s✉❧t❛❞♦s ♣♦ss✁✙✈❡✐s ✭✐✳❡✳ ❞❡ r❡s✉❧t❛❞♦s

❡❧❡♠❡♥t❛r❡s ❡ ✐♥❞✐✈✐s✁✙✈❡✐s✮ ❞❡ss❛ ❡①♣❡r✐ê♥❝✐❛ ❡ ✉♠ ♦✉tr♦ ❝♦♥❥✉♥t♦ q✉❡ r❡♣r❡s❡♥t❡ t♦❞♦s ♦s ❛❝♦♥t❡❝✐✲ ♠❡♥t♦s ♣♦ss✁✙✈❡✐s ❞❛ ❡①♣❡r✐ê♥❝✐❛✳ ▼❛✐s ❝♦♥❝r❡t❛♠❡♥t❡✱ ❛ ♣r✁♦①✐♠❛ ❞❡✜♥✐çã♦ ✈❛✐ ❛♦ ❡♥❝♦♥tr♦ ❞❛q✉✐❧♦ q✉❡ ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ♥❡❝❡ss✐t❛♠♦s✳

❉❡✜♥✐çã♦ ✶✳✶✳✷ ❯♠❛ ❝❧❛ss❡F ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦Ωq✉❡ s❛t✐s❢❛ç❛✿ (σ1) Ω_{∈ F}✳

(σ2) ❙❡A_{∈ F} ❡♥tã♦Ω_\A_{∈ F}✳

(σ3) ❙❡An∈ F ♣❛r❛ n= 1,2, . . . ❡♥tã♦

∞ [

n=1

An∈ F✳

é ❝❤❛♠❛❞❛σ✲✁❛❧❣❡❜r❛ s♦❜r❡Ω✭♦✉ σ✲✁❛❧❣❡❜r❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✮✳

❉❛❞❛ ✉♠❛ ❝❧❛ss❡ C ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ Ω ❝❤❛♠❛♠♦s σ✲✁❛❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r C ❛

♠❡♥♦rσ✲✁❛❧❣❡❜r❛ s♦❜r❡ Ω q✉❡ ❝♦♥té♠_C ❡ ✐♥❞✐❝❛♠♦s ❡ss❡ ❢❛❝t♦ ❡s❝r❡✈❡♥❞♦σ(_C)✳ ◆♦t❡✲s❡✱ ❡♥tã♦✱ q✉❡

✶✳✶✳ ❚❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✸

❊①❡♠♣❧♦ ✶✳✶✳✶

❙❡ X ❢♦r ✉♠ ❡s♣❛ç♦ t♦♣♦❧✁♦❣✐❝♦✱ ❝❤❛♠❛♠♦s σ✲✁❛❧❣❡❜r❛ ❞♦s ❜♦r❡❧✐❛♥♦s ❞❡ X ❛ σ✲✁❛❧❣❡❜r❛ s♦❜r❡ X ❣❡r❛❞❛ ♣❡❧❛ ❝❧❛ss❡ ❞♦s s✉❜❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❞❡ X✳ ❊st❛ σ✲✁❛❧❣❡❜r❛ é r❡♣r❡s❡♥t❛❞❛ ✉s✉❛❧♠❡♥t❡ ♣♦r B(X) ❡ ♦s s❡✉s ❡❧❡♠❡♥t♦s tê♠ ♦ ♥♦♠❡ ❞❡ ❜♦r❡❧✐❛♥♦s ❞❡ X✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ✐♠♣♦rt❛♥t✁✙ss✐♠♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ X=❘n✳

❆ σ✲✁❛❧❣❡❜r❛ ❞♦s ❜♦r❡❧✐❛♥♦s ❞❡ ❘n _{é✱ s❡♠ ❞✁✉✈✐❞❛✱ ❛ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❡ t♦❞❛s ❛sσ✲✁❛❧❣❡❜r❛s ♠❛s} ♦s s❡✉s ❡❧❡♠❡♥t♦s ♣♦❞❡♠ s❡r s✉❜❝♦♥❥✉♥t♦s ❞❡ ❘n _{❜❛st❛♥t❡ ❝♦♠♣❧✐❝❛❞♦s✳ ◆❡st❡ s❡♥t✐❞♦✱ ❛ ❞❡✜♥✐çã♦ ❡} ♦ ❡①❡♠♣❧♦ q✉❡ s❡ s❡❣✉❡♠ tê♠ ♣♦r ♦❜❥❡❝t✐✈♦ ❝❧❛r✐✜❝❛r ✉♠ ♣♦✉❝♦ ♠❛✐s ❡st❡ ❛ss✉♥t♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✸ ❙❡♥❞♦Ω✉♠ ❝♦♥❥✉♥t♦ ❝❤❛♠❛♠♦sπ✲s✐st❡♠❛ s♦❜r❡Ω❛ t♦❞❛ ❛ ❢❛♠✁✙❧✐❛I ❞❡ s✉❜❝♦♥✲ ❥✉♥t♦s ❞❡ Ω❢❡❝❤❛❞❛ ♣❛r❛ ❛ ✐♥t❡rs❡❝çã♦ ✜♥✐t❛✱

A, B∈ I =⇒ A∩B ∈ I.

❊①❡♠♣❧♦ ✶✳✶✳✷

❯♠❛ ❡①❡♠♣❧♦ t✁✙♣✐❝♦ ❞❡ ✉♠π✲s✐st❡♠❛ é ❛ ❝❧❛ss❡ ❢♦r♠❛❞❛ ♣♦r t♦❞♦s ♦s r❡❝t✂❛♥❣✉❧♦s ❞❡ ❘n ❝♦♠ ❛ ❢♦r♠❛✱

(x1, . . . , xn) : x1 6a1, . . . , xn6an, ai∈❘, i= 1, . . . , n

❡ ❞❡s✐❣♥❛❞❛ ❤❛❜✐t✉❛❧♠❡♥t❡ ♣♦rπ(❘n)✳ ❆❧é♠ ❞✐ss♦✱ ❛ σ✲✁❛❧❣❡❜r❛ ❞♦s ❜♦r❡❧✐❛♥♦s ❞❡ ❘n ❝♦✐♥❝✐❞❡ ❝♦♠ ❛

σ✲✁❛❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r π(❘n)✱ ♦✉ s❡❥❛✱_B(❘n) =σ(π(❘n))✳

❆ ✐❞❡✐❛ ❞❡ ❛ss♦❝✐❛r✱ ❛ ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛✱ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣♦ss✁✙✈❡✐s ❡ ❛ r❡s♣❡❝t✐✈❛ ❝❧❛ss❡ ❞❡ s✉❜❝♦♥❥✉♥t♦s q✉❡ ✐❞❡♥t✐✜q✉❡ t♦❞♦s ♦s ❛❝♦♥t❡❝✐♠❡♥t♦s ♣♦ss✁✙✈❡✐s ❞❛ ❡①♣❡r✐ê♥❝✐❛ ♣♦❞❡ ❛❣♦r❛ s❡r ❢♦r♠❛❧✐③❛❞❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✹ ❖ ❡s♣❛ç♦ ❛♠♦str❛❧ ❞❡ ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛ é ✉♠ ♣❛r (Ω,_F) ♦♥❞❡ Ω é ♦

❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦ss✁✙✈❡✐s r❡s✉❧t❛❞♦s ❞❛ ❡①♣❡r✐ê♥❝✐❛ ❡F ✉♠❛σ✲✁❛❧❣❡❜r❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✳

❖s ❡❧❡♠❡♥t♦s ❞❡ F ❞❡s✐❣♥❛♠✲s❡ ♣♦r ❛❝♦♥t❡❝✐♠❡♥t♦s❀ ❡♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ ω _∈ Ω ❝❤❛♠❛✲s❡

❛❝♦♥t❡❝✐♠❡♥t♦ ❡❧❡♠❡♥t❛r ❛{ω}✳ ◆❛ ❧✐♥❣✉❛❣❡♠ ❞❛ t❡♦r✐❛ ❞❛ ♠❡❞✐❞❛✱ ♦ ♣❛r(Ω,F)❞❛ ❉❡✜♥✐çã♦ ✶✳✶✳✹

❞❡s✐❣♥❛✲s❡ ♣♦r ❡s♣❛ç♦ ♠❡♥s✉r✁❛✈❡❧ ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡F sã♦ ❝❤❛♠❛❞♦s ❞❡F✲♠❡♥s✉r✁❛✈❡✐s ✭♦✉ s✐♠✲

♣❧❡s♠❡♥t❡ ♠❡♥s✉r✁❛✈❡✐s✮✳

❙❡♥❞♦ Ω ✉♠ ❝♦♥❥✉♥t♦✱ ❝❤❛♠❛♠♦s ❢✉♥çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❛ q✉❛❧q✉❡r ❢✉♥çã♦ ❝♦♠ ✈❛❧♦r❡s r❡❛✐s

❞❡✜♥✐❞❛ ♥✉♠❛ ❝❧❛ss❡ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✳ ❆ ♣r✁♦①✐♠❛ ❞❡✜♥✐çã♦✱ ❛♣r❡s❡♥t❛ ♦s três ❛①✐♦♠❛s ❢✉♥❞❛✲

♠❡♥t❛✐s ♣❛r❛ ❛ ♥♦çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✶✳✶✳✺ ❙❡❥❛ (Ω,F) ✉♠ ❡s♣❛ç♦ ❛♠♦str❛❧✳ ❯♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ é ✉♠❛ ❢✉♥çã♦

❞❡ ❝♦♥❥✉♥t♦ P(_·)❞❡✜♥✐❞❛ ❡♠ _F q✉❡ s❛t✐s❢❛③ ♦s ❛①✐♦♠❛s✿

✹ ✶✳ Pr❡❧✐♠✐♥❛r❡s

✭P✷✮ ❙❡{An, n>1}é ✉♠❛ s✉❝❡ssã♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡F ♠✉t✉❛♠❡♥t❡ ❡①❝❧✉s✐✈♦s ✭✐st♦ é✱Ai∩Aj =∅ s❡i₆=j✮ ❡♥tã♦ P

∞ [

n=1 An

! =

∞ X

n=1

P(An)✳ ✭P✸✮ P(Ω) = 1✳

❖ t❡r♠♦ ✧♠❡❞✐❞❛✧♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ❛❞✈é♠ ❞♦ ❢❛❝t♦ ❞❡✱ ♥♦ q✉❛❞r♦ ❞❛ t❡♦r✐❛ ❞❛ ♠❡❞✐❞❛✱ ❞❛❞♦ ✉♠ ❡s♣❛ç♦ ♠❡♥s✉r✁❛✈❡❧ (Ω,_F) ❝❤❛♠❛✲s❡ ♠❡❞✐❞❛ s♦❜r❡ (Ω,_F) ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ♠: _{F →}[0,_∞] q✉❡

✈❡r✐✜❝❛ ♠(∅) = 0 ❡ ♦ ❛①✐♦♠❛ ✭P✷✮✳ ❖ tr✐♣❧♦ (Ω,F,♠) ❞✐③✲s❡ ❡♥tã♦ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛✳

❊①❡♠♣❧♦ ✶✳✶✳✸

❙❡♥❞♦X ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r✱(X,F)✉♠ ❡s♣❛ç♦ ♠❡♥s✉r✁❛✈❡❧ ❡a✉♠ ❡❧❡♠❡♥t♦ ✜①❛❞♦ ❞❡X♣♦❞❡♠♦s ❞❡✜♥✐r ❡♠ F ✉♠❛ ♠❡❞✐❞❛ ♠ ♣♦r✿ ♠(A) = 0 s❡ a /_∈ A ❡ ♠(A) = 1 s❡ a _∈A✱ ♣❛r❛ t♦❞♦ ♦ A ∈ F✳ ❊st❛ é ❛ ♠❡❞✐❞❛ ❞❡ ❉✐r❛❝ r❡❧❛t✐✈❛ ❛♦ ♣♦♥t♦a✳

❊①❡♠♣❧♦ ✶✳✶✳✹

❯♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ ❞❡ ❣r❛♥❞❡ r❡❧❡✈♦ é(❘,_B(❘),▲❡❜) ♦♥❞❡ ▲❡❜ ❞❡s✐❣♥❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡

s♦❜r❡ ❘✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❝❧❛ss❡S ❞❡ ♣❛rt❡s ❞❡ ❘ ❝♦♥st✐t✉✁✙❞❛ ♣♦r t♦❞♦s ♦s ✐♥t❡r✈❛❧♦s s❡♠✐✲❛❜❡rt♦s ❞❛ ❢♦r♠❛[a, b[ (a6b)✶ ❡♥tã♦ ▲❡❜:_B(❘)_−→[0,_∞]é ❛ ✭✁✉♥✐❝❛✮ ♠❡❞✐❞❛ s♦❜r❡ ❛ σ✲✁❛❧❣❡❜r❛ ❞♦s ❜♦r❡❧✐❛♥♦s ❞❡ ❘ q✉❡ ♣r♦❧♦♥❣❛ ▲❡❜:_{S −→}[0,_∞]❞❛❞❛ ♣♦r ▲❡❜([a, b[) =b₋a✳

◆❡st❡ ♠♦♠❡♥t♦✱ ❡st❛♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❛♥✉♥❝✐❛r ❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♠♦❞❡❧♦ ♠❛t❡♠✁❛t✐❝♦ ♣❛r❛ ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛✱ ♦✉ s❡❥❛✱ ♦ ♠♦❞❡❧♦ ♣r♦❜❛❜✐❧✁✙st✐❝♦ q✉❡ s❡r✈✐r✁❛ ❞❡ ❜❛s❡ ♣❛r❛ t♦❞♦s ♦s ❡st✉❞♦s q✉❡ ❡❢❡❝t✉❛r♠♦s ❡ q✉❡ é ❝♦♥st✐t✉✁✙❞♦ ♣♦r✿

▼✶✳ ❯♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦Ω✱ ❞❡ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣♦ss✁✙✈❡✐s✳

▼✷✳ ❯♠❛ σ✲✁❛❧❣❡❜r❛F s♦❜r❡Ω❞❡ ❛❝♦♥t❡❝✐♠❡♥t♦s ❛❧❡❛t✁♦r✐♦s✳

▼✸✳ ❯♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ P ❞❡✜♥✐❞❛ ❡♠F✳ ❊♥✜♠✱ ♥✉♠ ❝♦♥❝❡✐t♦ ♠❛t❡♠✁❛t✐❝♦ ❛❜str❛❝t♦ t❡♠✲s❡ ❛✱

❉❡✜♥✐çã♦ ✶✳✶✳✻ ❯♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ é ✉♠ tr✐♣❧♦(Ω,F,P) ♦♥❞❡ Ωé ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲

✈❛③✐♦✱ F é ✉♠❛ σ✲✁❛❧❣❡❜r❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ❡ P é ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡✜♥✐❞❛ ❡♠

F✳

◆✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ (Ω,F,P) ❞✐③❡♠♦s q✉❡ ✉♠ ❛❝♦♥t❡❝✐♠❡♥t♦ A ∈ F ♦❝♦rr❡ q✉❛s❡ ❝❡rt❛♠❡♥t❡ ✭❛❜r❡✈✐❛❞❛♠❡♥t❡ q✳❝✳✮ s❡ P(A) = 1✳

✶_{❯♠❛ ❝❧❛ss❡C ❞❡ ♣❛rt❡s ❞❡ ✉♠ ❝♦♥❥✉♥t♦Ω❞✐③✲s❡ ✉♠ s❡♠✐✲❛♥❡❧ s❡ s❡ ✈❡r✐✜❝❛r❡♠ ❛s ♣r♦♣r✐❡❞❛❞❡s s❡❣✉✐♥t❡s✿}

✭❙✶✮ ∅ ∈ C✳

✭❙✷✮ ❙❡A_{∈ C} ❡B_{∈ C}❡♥tã♦A_∩B_{∈ C}✳

✭❙✸✮ ❙❡ A _{∈ C} ❡ B _{∈ C} ❡♥tã♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ✁✙♥❞✐❝❡sI ❡ ✉♠❛ ❢❛♠✁✙❧✐❛ {Ai, i ∈ I} ❞❡ s✉❜❝♦♥❥✉♥t♦s

♣❡rt❡♥❝❡♥t❡s ❛S ❞✐s❥✉♥t♦s ❞♦✐s ❛ ❞♦✐s✱ t❛❧ q✉❡A₋B=S_i∈IAi✳

✶✳✶✳ ❚❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✺

❙❡❥❛{An, n>1} ✉♠❛ s✉❝❡ssã♦ ❞❡ ❛❝♦♥t❡❝✐♠❡♥t♦s ♥✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡(Ω,F,P)✳ ❉❡✜✲ ♥✐♠♦s ❧✐♠✐t❡ s✉♣❡r✐♦r ❞❛ s✉❝❡ssã♦ {An, n>1}♣♦r✱

lim sup

n→∞ An= ∞ \

n=1

∞ [

m=n Am.

❖ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ❞❛ s✉❝❡ssã♦ {An, n>1} é ❞❡✜♥✐❞♦ ♣♦r✱

lim inf

n→∞ An= ∞ [

n=1

∞ \

m=n Am.

❖ ❛❝♦♥t❡❝✐♠❡♥t♦lim supAné ♦ ❛❝♦♥t❡❝✐♠❡♥t♦ ✧♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ♥✁✉♠❡r♦ ✐♥✜♥✐t♦ ❞♦sAn✧✱ ♥♦ s❡♥t✐❞♦ ❡♠ q✉❡ω ∈lim supAn s❡ ❡ s✁♦ s❡ω ♣❡rt❡♥❝❡ ❛ ✉♠ ♥✁✉♠❡r♦ ✐♥✜♥✐t♦ ❞♦sAn✳ ❯♠❛ ♥♦t❛çã♦ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ lim supAn é{An, ✐♥✜♥✐t❛s ✈❡③❡s}✳ ❖ ❛❝♦♥t❡❝✐♠❡♥t♦ lim infAn t❛♠❜é♠ t❡♠ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ✐♥t✉✐t✐✈❛✿ é ♦ ❛❝♦♥t❡❝✐♠❡♥t♦ ✧♦❝♦rrê♥❝✐❛ ❞❡ An ♣❛r❛ t♦❞♦ ♦ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✧✳ ❆ r❛③ã♦ ♣r❡♥❞❡✲s❡ ❝♦♠ ♦ ❢❛❝t♦ ❞❡ ω _∈ lim infAn s❡ ❡ s✁♦ s❡ ω ∈ Am ♣❛r❛ t♦❞♦ ♦ m s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ✭m>n0(ω)✮✳ ❊♠ ❝♦♥tr❛♣♦♥t♦ ❝♦♠ ❛ ♥♦t❛çã♦{An, ✐♥✜♥✐t❛s ✈❡③❡s}❛❧❣✉♥s ❛✉t♦r❡s ✐♥❞✐❝❛♠lim infAn ♣♦r {An, q✉❛s❡ ❝❡rt❛♠❡♥t❡} ✭✈❡r ❬✾✶❪✮✳ ◆♦ s❡❣✉✐♠❡♥t♦ ❞♦ q✉❡ ❛❝❛❜✁❛♠♦s ❞❡ ❡①♣♦r ✈❡♠ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ t❡♦r❡♠❛ ❞❡ ❇♦r❡❧✲❈❛♥t❡❧❧✐✱ ✉♠ r❡s✉❧t❛❞♦ s✐♥❡ q✉❛ ♥♦♥ ❡♠ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡ q✉❡ ❝♦♥st✐t✉✐ ✉♠ ✐♥str✉♠❡♥t♦ ♣♦❞❡r♦s♦ ♥❛s ❧❡✐s ❞❡ ❣r❛♥❞❡s ♥✁✉♠❡r♦s✳ ❆♦ ❧♦♥❣♦ ❞❡st❛ s❡❝çã♦✱ r❡♠❡t❡r❡♠♦s ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❛♥✉♥❝✐❛❞♦s ♣❛r❛ ♦s ❧✐✈r♦s ❣❡♥❡r❛❧✐st❛s s♦❜r❡ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❝✐t❛❞♦s ♥❛ ❜✐❜❧✐♦❣r❛✜❛✳

❚❡♦r❡♠❛ ✶✳✶✳✶ ✭♣r✐♠❡✐r♦ ❧❡♠❛ ❞❡ ❇♦r❡❧✲❈❛♥t❡❧❧✐✮ ❙❡❥❛ {An, n>1} ✉♠❛ s✉❝❡ssã♦ ❞❡ ❛❝♦♥t❡✲ ❝✐♠❡♥t♦s ♥✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ (Ω,_F,P) t❛❧ q✉❡ P∞_n=1P(An)<∞ ❡♥tã♦✱

P

lim sup

n→∞

An

= 0.

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✉❧t❛r ❬✾✶❪✱ ♣✁❛❣✐♥❛27✳

❱❡❥❛♠♦s ❛❣♦r❛ ❝♦♠♦ ❞❡✜♥✐r ✈❛r✐✁❛✈❡❧ ❛❧❡❛t✁♦r✐❛✳ ◆❛ ♣r✁❛t✐❝❛✱ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ ❛♠♦str❛❧ ♣♦❞❡♠ ❝♦✐s❛s ♠✉✐t♦ ❝♦♥❝r❡t❛s t❛✐s ❝♦♠♦ ❛s ❢❛❝❡s ❞❡ ✉♠ ❞❛❞♦✱ ❛s ♠♦❧é❝✉❧❛s ❞❡ ✉♠ ❣✁❛s ♦✉ ❛té s❡r❡s ❤✉♠❛♥♦s✳ ❈♦♥t✉❞♦✱ ♥ã♦ sã♦ ♦s ♣r✁♦♣r✐♦s ❡❧❡♠❡♥t♦s ❞♦ ❡s♣❛ç♦ ❛♠♦str❛❧ q✉❡ ♥♦s ✐♥t❡r❡ss❛♠ ♠❛s s✐♠ ✈❛❧♦r❡s ♥✉♠ér✐❝♦s q✉❡ ❛ ❡❧❡s ❡stã♦ ❛ss♦❝✐❛❞♦s✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ s❡r ❤✉♠❛♥♦ ♣♦❞❡♠♦s ♠❡❞✐r ❝❡rt❛s ❝❛r❛❝t❡r✁✙st✐❝❛s ❢✁✙s✐❝❛s t❛✐s ❝♦♠♦ ❛ ✐❞❛❞❡✱ ♦ ♣❡s♦ ❡ ❛ ❛❧t✉r❛ ♦✉ ❛✐♥❞❛ ♦✉tr❛s ❝❛r❛❝t❡r✁✙st✐❝❛s t❛✐s ❝♦♠♦ ♦ r❡♥❞✐♠❡♥t♦ ❛♥✉❛❧✱ ♦ ♥✁✉♠❡r♦ ❞❡ ❛♥♦s ❞❡ ❡s❝♦❧❛r✐❞❛❞❡✳ ■♥❢♦r♠❛❧♠❡♥t❡✱ ✉♠❛ ✈❛r✐✁❛✈❡❧ ❛❧❡❛t✁♦r✐❛ s❡r✁❛ ❡♥tã♦ ✉♠❛ ❝❛r❛❝t❡r✁✙st✐❝❛ ♥✉♠ér✐❝❛ ❞♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛❧❡❛t✁♦r✐❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✼ ❉❛❞♦ ✉♠ ❡s♣❛ç♦ ❛♠♦str❛❧ (Ω,F) ❝❤❛♠❛♠♦s ✈❛r✐✁❛✈❡❧ ❛❧❡❛t✁♦r✐❛ ✭❛❜r❡✈✐❛❞❛♠❡♥t❡

✻ ✶✳ Pr❡❧✐♠✐♥❛r❡s

◆♦ ✂❛♠❜✐t♦ ❞❛ t❡♦r✐❛ ❞❛ ♠❡❞✐❞❛✱ s❡ (Ω,_F) ❡ (Σ,_G) ❢♦r❡♠ ❡s♣❛ç♦s ♠❡♥s✉r✁❛✈❡✐s ❞✐③✲s❡ q✉❡ ✉♠❛

❢✉♥çã♦ ↕: Ω _→ Σ é F✲♠❡♥s✉r✁❛✈❡❧ ✭♦✉ ❛♣❡♥❛s ♠❡♥s✉r✁❛✈❡❧✮ s❡ ↕−1_{(A) = ω ∈ Ω :} ↕_{(ω) ∈ A ∈ F}

♣❛r❛ ❝❛❞❛ A _{∈ G} ✭❢❛③❡♥❞♦ ❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛ ❉❡✜♥✐çã♦ ✶✳✶✳✼ ❝♦♥❝❧✉✐✲s❡ q✉❡ ✉♠❛ ✈✳❛✳ ♥ã♦ é ♠❛✐s ❞♦ q✉❡ ✉♠❛ ❢✉♥çã♦ F✲♠❡♥s✉r✁❛✈❡❧ ❝✉❥♦ ♦ ❡s♣❛ç♦ ♠❡♥s✉r✁❛✈❡❧(Σ,_G)é (❘,_B(❘))✮✳

❊①❡♠♣❧♦ ✶✳✶✳✺

❙❡ X é ✉♠❛ ✈✳❛✳ ❞❡✜♥✐❞❛ ❡♠ (Ω,_F) ❡ ✩:❘ _→ ❘ ✉♠❛ ❢✉♥çã♦ ❜♦r❡❧✐❛♥❛ ✭✐✳❡✳ _B(❘)✲♠❡♥s✉r✁❛✈❡❧

q✉❛♥❞♦(Ω,_F) = (❘,_B(❘)) = (Σ,_G)✮ ❡♥tã♦ ✩(X) é ❛✐♥❞❛ ✉♠❛ ✈✳❛✳ ✭❡♠ ♣❛rt✐❝✉❧❛r✱ s❡ ✩ ❢♦r ❝♦♥t✁✙♥✉❛

♦✉ ♠♦♥✁♦t♦♥❛ ❡♥tã♦ é ❜♦r❡❧✐❛♥❛✮✳

❙❡ {Yj, j ∈ J} ❢♦r ✉♠❛ ❢❛♠✁✙❧✐❛ ♥ã♦✲✈❛③✐❛ ❞❡ ✈✳❛✳✬s s♦❜r❡ (Ω,F) t❡♠ ❧✉❣❛r ✉♠❛ σ✲✁❛❧❣❡❜r❛ ❞❡ ❡♥♦r♠❡ r❡❧❡✈✂❛♥❝✐❛✿ ❛ σ✲✁❛❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉♥t♦sω: Yj(ω)∈B ✱ B ∈ B(❘)✱ j ∈J ❞❡s✐❣♥❛❞❛ ❤❛❜✐t✉❛❧♠❡♥t❡ ♣♦rσ(Yj, j ∈J)✳ ❈❧❛r❛♠❡♥t❡✱ σ(Yj, j ∈J)⊂ F s❡♥❞♦ t❛♠❜é♠ ❛ ♠❡♥♦r σ✲✁❛❧❣❡❜r❛ r❡❧❛t✐✈❛♠❡♥t❡ ❛ q✉❛❧ t♦❞♦s ♦sYj✱j ∈J sã♦ ♠❡♥s✉r✁❛✈❡✐s✳

◆♦ ❝♦♥❝❡✐t♦ ❞❡ ✈✳❛✳ ❞❡✜♥✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♥ã♦ ❡♥❝♦♥tr❛♠♦s q✉❛❧q✉❡r r❡❢❡rê♥❝✐❛ ❛ ✉♠❛ ❡✈❡♥t✉❛❧ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ P ❞❡✜♥✐❞❛ ♥♦ ❡s♣❛ç♦ ❛♠♦str❛❧✳ P♦ré♠✱ ♥❛ ♣r✁❛t✐❝❛✱ ❛s ✈❛r✐✁❛✈❡✐s ❛❧❡❛t✁♦r✐❛s q✉❡ ✐♥t❡r❡ss❛♠ ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ❡♥❝♦♥tr❛♠✲s❡ ❞❡✜♥✐❞❛s ♥✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ ❡stã♦✱ ❝♦♠♦ ✐r❡♠♦s ✈❡r✱ ✐♥t✐♠❛♠❡♥t❡ ❧✐❣❛❞❛s ❛ ✉♠❛ ❝❧❛ss❡ ❡s♣❡❝✐❛❧ ❞❡ ❢✉♥çõ❡s r❡❛✐s ❞❡ ✈❛r✐✁❛✈❡❧ r❡❛❧✱ ❛ s❡r ✐♥tr♦❞✉③✐❞❛ ❥✁❛ ❞❡ s❡❣✉✐❞❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✽ ❯♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐✁❛✈❡❧ r❡❛❧F ❞❡✜♥✐❞❛ ❡♠ t♦❞♦ ♦ ❘ q✉❡ s❡❥❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡✷✱ ❝♦♥t✁✙♥✉❛ ❛ ❞✐r❡✐t❛✸ _{❡ s❛t✐s❢❛ç❛✱}

lim

x→−∞F(x) = 0 ❡ x→lim+∞F(x) = 1

é ❞❡s✐❣♥❛❞❛ ♣♦r ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ✭❛❜r❡✈✐❛❞❛♠❡♥t❡ ❢✳❞✳✮✳ ❖ s✉♣♦rt❡ ❞❡ ✉♠❛ ❢✳❞✳ ❛r❜✐tr✁❛r✐❛F é ♦ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✱

❙✉♣♣(F) =x: F(x+ε)₋F(x₋ε)>0, _∀ε >0

✭✈❡r ❬✶✽❪ ♦✉ ❬✶✾❪✮✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ ❙✉♣♣(F) sã♦ ❝❤❛♠❛❞♦s ♣♦♥t♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡F✳ ❯♠❛ ❢✳❞✳

F ❞✐③✲s❡ ❞❡❣❡♥❡r❛❞❛ s❡ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠ ✁✉♥✐❝♦ ♣♦♥t♦ ❞❡ ❝r❡s❝✐♠❡♥t♦❀ ❝❛s♦ ❝♦♥tr✁❛r✐♦ ❞✐③❡♠♦s q✉❡ F é ♥ã♦✲❞❡❣❡♥❡r❛❞❛✳ ❱❛♠♦s ❛❣♦r❛ ✐♥tr♦❞✉③✐r ❛❧❣✉♠❛s ♥♦t❛çõ❡s q✉❡ ✉s❛r❡♠♦s ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ t❡①t♦✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ❛ ❝❛✉❞❛ ❞❡ ✉♠❛ ❢✳❞✳ F ♣♦r✱

F(x) = 1₋F(x), x>0.

✷_{❘❡❝♦r❞❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ F ❞❡✜♥✐❞❛ ❡♠D ⊂❘ é ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ✭r❡s♣✳ ♥ã♦✲❝r❡s❝❡♥t❡✮ s❡ ❞❛❞♦sx, y ∈ D✱}

x < y =_⇒ F(x) 6F(y) ✭r❡s♣✳ x, y_∈D✱ x < y =_⇒ F(x) >F(y)✮✳ ❊♥✜♠✱ ✉♠❛ ❢✉♥çã♦ F ❞❡✜♥✐❞❛ ❡♠D _⊂❘ é

❝r❡s❝❡♥t❡ ✭r❡s♣✳ ❞❡❝r❡s❝❡♥t❡✮ s❡ ❞❛❞♦sx, y_∈D✱x < y =_⇒ F(x)< F(y)✭r❡s♣✳ x, y_∈D✱x < y =_⇒ F(x)> F(y)✮✳

✸_{▲❡♠❜r❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦F ❞❡✜♥✐❞❛ ❡♠ t♦❞♦ ♦ ❘ é ❝♦♥t✁✙♥✉❛ ❛ ❞✐r❡✐t❛ ❡♠ ❘ s❡ lim}

y→x y>x