TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD

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50 Teorías de la Medida y de la Probabilidad Colección manuales uex - 57  E. E. MANUALES UEX  FSEF o n d o S o c i a l E u ro p e o Edita Universidad de Extremadura. Servicio de PublicacionesC./ Caldereros, 2 - Planta 2ª - 10071 Cáceres (España) Telf. ISSN 1135-870-XISBN 978-84-691-6411-2 Depósito Legal M-45.182-2008Edición electrónica: Pedro Cid, S.A. A María José y Carlos A mis padres ❮♥❞✐❝❡ ❣❡♥❡r❛❧Pró❧♦❣♦✳ ✼ ✶✳ ❊s♣❛❝✐♦s ▼❡❞✐❜❧❡s ✶✶ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳ ❋✉♥❝✐♦♥❡s ▼❡❞✐❜❧❡s✳ ❱❛r✐❛❜❧❡s ❆❧❡❛t♦r✐❛s ✶✾Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳ ❊s♣❛❝✐♦s ❞❡ ▼❡❞✐❞❛✳ ❊s♣❛❝✐♦s ❞❡ Pr♦❜❛❜✐❧✐❞❛❞ ✷✾ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳ ■♥t❡❣r❛❧✳ ❊s♣❡r❛♥③❛ ✹✸ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳ ❙✉♠❛ ❞❡ ▼❡❞✐❞❛s✳ ▼❡❞✐❞❛ ■♠❛❣❡♥✳❉✐str✐❜✉❝✐♦♥❡s ❞❡ Pr♦❜❛❜✐❧✐❞❛❞ ✺✼ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✻✳ ▼❡❞✐❞❛s ❉❡✜♥✐❞❛s ♣♦r ❉❡♥s✐❞❛❞❡s✳❚❡♦r❡♠❛ ❞❡ ❈❛♠❜✐♦ ❞❡ ❱❛r✐❛❜❧❡s ✻✼ M A N U A LE S U EX ❮◆❉■❈❊ ●❊◆❊❘❆▲❮◆❉■❈❊ ●❊◆❊❘❆▲ ❉✐str✐❜✉❝✐ó♥ ❈♦♥❥✉♥t❛ ❞❡ ❱✳❆✳ ❉✐str✐❜✉❝✐♦♥❡s ❞✐s❝r❡t❛s ✉♥✐✈❛r✐❛♥t❡s ❡♥ ❚❡♦rí❛ ❞❡ ❧❛ Pr♦❜❛❜✐❧✐❞❛❞ ② ❡♥ ❊st❛❞íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✸AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ✶✽✸ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✵ ❆P➱◆❉■❈❊❙ ✶✼✶ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✶✹✳ ❉❡s✐❣✉❛❧❞❛❞ ❞❡ ❏❡♥s❡♥✳Pr♦❜❧❡♠❛ ●❡♥❡r❛❧ ❞❡ ❘❡❣r❡s✐ó♥ ✶✺✼ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷ ✶✷✳ ❉❡✜♥✐❝✐ó♥ ❞❡ ❊s♣❡r❛♥③❛ ❈♦♥❞✐❝✐♦♥❛❧ ✶✹✸Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✹ ✶✸✳ ❉✐str✐❜✉❝✐ó♥ ❈♦♥❞✐❝✐♦♥❛❧ ② Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✼✳ ▼❡❞✐❞❛ Pr♦❞✉❝t♦✳ ▼❡❞✐❞❛s ❞❡ ❚r❛♥s✐❝✐ó♥ ✼✺Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✽✳ ■♥❞❡♣❡♥❞❡♥❝✐❛✳ ❈♦♥✈♦❧✉❝✐ó♥ ✶✸✺ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✶✶✳ ❚❡♦r❡♠❛ ❞❡ ❘❛❞♦♥✲◆✐❦♦❞②♠ ✶✶✺ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶ ✶✵✳ ▼✉❡str❛s ✶✵✺ Pr♦❜❧❡♠❛s ❞❡❧ ❈❛♣ít✉❧♦ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✾✳ ❋✉♥❝✐ó♥ ❈❛r❛❝t❡ríst✐❝❛ ✽✼ M A N U A LE S U EX ❮◆❉■❈❊ ●❊◆❊❘❆▲❮◆❉■❈❊ ●❊◆❊❘❆▲ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD❉✐str✐❜✉❝✐♦♥❡s ❝♦♥t✐♥✉❛s ✉♥✐✈❛r✐❛♥t❡s ❡♥ ❚❡♦rí❛ ❞❡ ❧❛ Pr♦❜❛❜✐❧✐❞❛❞ ② ❡♥ ❊st❛❞íst✐❝❛ ✭■✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✹❉✐str✐❜✉❝✐♦♥❡s ❝♦♥t✐♥✉❛s ✉♥✐✈❛r✐❛♥t❡s ❡♥ ❚❡♦rí❛ ❞❡ ❧❛ Pr♦❜❛❜✐❧✐❞❛❞ ② ❡♥ ❊st❛❞íst✐❝❛ ✭■■✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✺❉✐str✐❜✉❝✐♦♥❡s ♠✉❧t✐✈❛r✐❛♥t❡s ❡♥ ❚❡♦rí❛ ❞❡ ❧❛ Pr♦❜❛❜✐❧✐❞❛❞ ② ❡♥ ❊st❛❞íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✻ ❮◆❉■❈❊ ❆▲❋❆❇➱❚■❈❖ ✶✽✼ EX U S LE A U N A M Pró❧♦❣♦ ❊st❡ ♠❛♥✉❛❧ s♦❜r❡ ❧❛s t❡♦rí❛s ❞❡ ❧❛ ▼❡❞✐❞❛ ② ❞❡ ❧❛ Pr♦❜❛❜✐❧✐❞❛❞ ❤❛ s✐❞♦ ❡❧❛❜♦r❛❞♦❛ ♣❛rt✐r ❞❡ ❧♦s ❛♣✉♥t❡s ❞❡ ❝❧❛s❡ ❞❡ ✉♥❛ ❛s✐❣♥❛t✉r❛ ❤♦♠ó♥✐♠❛ ❡♥ ❡st✉❞✐♦s ❞❡ ❊st❛❞íst✐❝❛❡♥ ❧❛ ❯♥✐✈❡rs✐❞❛❞ ❞❡ ❊①tr❡♠❛❞✉r❛ ② ♣r❡t❡♥❞❡✱ ❛♥t❡s q✉❡ ❝♦♥✈❡rt✐rs❡ ❡♥ ✉♥❛ ♦❜r❛❡①❤❛✉st✐✈❛ ② ❛✉t♦s✉✜❝✐❡♥t❡ s♦❜r❡ ▼❡❞✐❞❛ ② Pr♦❜❛❜✐❧✐❞❛❞✱ s❡r ❞❡ ✉t✐❧✐❞❛❞ ❡♥ ❝❧❛s❡✳ ❙✉ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❝♦♥s✐st❡ ❡♥ ♣r❡s❡♥t❛r ❧❛ ❚❡♦rí❛ ❞❡ ❧❛ Pr♦❜❛❜✐❧✐❞❛❞ ❞❡s❞❡ ✉♥♣✉♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝❛♠❡♥t❡ r✐❣✉r♦s♦ ❛ ♣❛rt✐r ❞❡❧ tr♦♥❝♦ ❝♦♠ú♥ q✉❡ ❝♦♠♣❛rt❡❝♦♥ ❧❛ ❚❡♦rí❛ ❞❡ ❧❛ ▼❡❞✐❞❛✱ t❛❧ ② ❝♦♠♦ ♥♦s ❡♥s❡ñó ❤❛❝❡ ✼✺ ❛ñ♦s ❆♥❞ré✐ ◆✐❦♦❧á②❡✲✈✐❝❤ ❑♦❧♠♦❣ór♦✈✱ ♠❛t❡♠át✐❝♦ r✉s♦ q✉❡ ❝♦♥s✐❣✉✐ó ❝♦♥✈❡♥❝❡r♥♦s ❞❡ q✉❡ ♠❡❞✐r ár❡❛s♦ ✈♦❧ú♠❡♥❡s ② ❝❛❧❝✉❧❛r ♣r♦❜❛❜✐❧✐❞❛❞❡s ♥♦ s♦♥ ♣r♦❜❧❡♠❛s t❛♥ ❞✐st✐♥t♦s ❝♦♠♦ ♣✉❞✐❡✲ r❛♥ ♣❛r❡❝❡r ❡♥ ♣r✐♥❝✐♣✐♦✳ P❡r♠ít❛s❡♥♦s r❡♣r♦❞✉❝✐r ❛q✉í ♣❛rt❡ ❞❡❧ Pró❧♦❣♦ ❞❡ ◆♦❣❛❧❡s✭✶✾✾✽✮✿ ✧✳✳✳ ▲❛s tr❛②❡❝t♦r✐❛s ♣❛r❛❧❡❧❛s q✉❡ ❤✐stór✐❝❛♠❡♥t❡ s✐❣✉✐❡r♦♥ ❧❛ ❚❡♦rí❛ ❞❡ ❧❛ ▼❡✲❞✐❞❛ ② ❡❧ ❈á❧❝✉❧♦ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡s ❤❛st❛ q✉❡✱ ❡♥ ✶✾✸✸✱ ❆✳◆✳ ❑♦❧♠♦❣♦r♦✈ ✜❥ó ✉♥❛❜❛s❡ ❛①✐♦♠át✐❝❛ ❝♦♠ú♥ ♣❛r❛ ❛♠❜❛s t❡♦rí❛s✱ ❥✉st✐✜❝❛ ❧❛ ❞♦❜❧❡ t❡r♠✐♥♦❧♦❣í❛ q✉❡✱ ♣❛r❛♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s ✐❞é♥t✐❝♦s✱ s❡ ✉s❛♥ ❡♥ ❧❛s ♠✐s♠❛s✳✳✳ ❊♥ ♣❛❧❛❜r❛s ❞❡ ❏✳▲✳ ❉♦♦❜✭✈é❛s❡ ❉♦♦❜✱ ❏✳▲✳ ✭✶✾✽✾✮✱ ✏❑♦❧♠♦❣♦r♦✈✬s ❡❛r❧② ✇♦r❦ ♦♥ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r② ❛♥❞ ❢♦✉♥✲ EX ❞❛t✐♦♥s✑✱ ❆♥♥✳ ♦❢ Pr♦❜✳✱ ❱♦❧✳ ✶✼✱ ◆♦✳ ✸✱ ♣✳ ✽✶✺✳✮✱ ✏t❤✐s ✐♥✢✉❡♥t✐❛❧ ♠♦♥♦❣r❛♣❤ ✖s❡ U S r❡✜❡r❡ ❛ ❑♦❧♠♦❣♦r♦✈✱ ❆✳◆✳ ✭✶✾✸✸✮✱ ●r✉♥❞❜❡❣r✐✛❡ ❞❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐tsr❡❝❤♥✉♥❣✱ LE ❙♣r✐♥❣❡r✳✖ tr❛♥s❢♦r♠❡❞ t❤❡ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s✱ ♠♦✈✐♥❣ ✐t ✐♥t♦ A U ♠❛t❤❡♠❛t✐❝s ❢r♦♠ ✐ts ♣r❡✈✐♦✉s st❛t❡ ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❝❛❧❝✉❧❛t✐♦♥s ✐♥s♣✐r❡❞ ❜② ❛ ✈❛✲ N A ❣✉❡ ♥♦♥♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥t❡①t✱ ❛ ❝♦♥t❡①t t❤♦✉❣❤t t♦ ❥✉st✐❢② t❤❡ ✉s❡ ♦❢ ❛ ❤❛❧❢✲❞❡✜♥❡❞ M ♣s❡✉❞♦♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥❝❡♣ts✳✑ ❊❧ ❧❡❝t♦r ♣✉❡❞❡ ❡♥❝♦♥tr❛r ❡♥ ✓❙❤✐r②❛❡✈✱ ❆✳◆✳ ✭✶✾✽✾✮✱ ❮◆❉■❈❊ ●❊◆❊❘❆▲❮◆❉■❈❊ ●❊◆❊❘❆▲ ✏❑♦❧♠♦❣♦r♦✈✲▲✐❢❡ ❛♥❞ ❝r❡❛t✐✈❡ ❛❝t✐✈✐t✐❡s✑✱ ❆♥♥✳ ♦❢ Pr♦❜✳✱ ❱♦❧✳ ✶✼✱ ◆♦✳ ✸✱ ♣✳ ✽✻✻✔ ✉♥❡st✉❞✐♦ ❞❡t❛❧❧❛❞♦ ❞❡❧ tr❛❜❛❥♦ ❞❡ ❑♦❧♠♦❣♦r♦✈ ② q✉❡✱ ❡♥ ♣❛rt✐❝✉❧❛r✱ r❡❝♦❣❡ ❧❛s ❞♦s ❝✐t❛s s✐❣✉✐❡♥t❡s✿ ■tô✿ ✏❍❛✈✐♥❣ r❡❛❞ ❑♦❧♠♦❣♦r♦✈✬s ❚❤❡ ❋♦✉♥❞❛t✐♦♥s ♦❢ Pr♦❜❛❜✐❧✐t② ❚❤❡♦r②✱ ■ ❜❡❝❛♠❡❝♦♥✈✐♥❝❡❞ t❤❛t ♣r♦❜❛❜✐❧✐t② t❤❡♦r② ❝♦✉❧❞ ❜❡ ❞❡✈❡❧♦♣❡❞ ✐♥ t❡r♠s ♦❢ ♠❡❛s✉r❡ t❤❡♦r②❛s r✐❣♦r♦✉s❧② ❛s ♦t❤❡r ✜❡❧❞s ♦❢ ♠❛t❤❡♠❛t✐❝s✳✑ ❑❛❝✿ ✭❞❡s❝r✐❜✐❡♥❞♦ s✉ ❝♦❧❛❜♦r❛❝✐ó♥ ❝♦♥ ❍✳ ❙t❡✐♥❤❛✉s✮ ✏❖✉r ✇♦r❦ ❜❡❣❛♥ ❛t ❛ t✐♠❡✇❤❡♥ ♣r♦❜❛❜✐❧✐t② t❤❡♦r② ✇❛s s❧♦✇❧② ❣❛✐♥✐♥❣ ❛❝❝❡♣t❛♥❝❡ ❛s ❛ r❡s♣❡❝t❛❜❧❡ ❜r❛♥❝❤♦❢ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s✳ ❚❤❡ t✉r♥❛❜♦✉t ❝❛♠❡ ❛s ❛ r❡s✉❧t ♦❢ ❛ ❜♦♦❦ ❜② t❤❡ ❣r❡❛t❙♦✈✐❡t ♠❛t❤❡♠❛t✐❝✐❛♥ ❆✳◆✳ ❑♦❧♠♦❣♦r♦✈ ♦♥ ❢♦✉♥❞❛t✐♦♥s ♦❢ ♣r♦❜❛❜✐❧✐t② t❤❡♦r②✱♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✸✸✳✑ ◆♦ ❝❛❜❡ ❞✉❞❛ ❞❡ q✉❡ ❡s ❡❧ tr❛❜❛❥♦ ❞❡ ❑♦❧♠♦❣♦r♦✈ ❡♥ ♣r♦❜❛❜✐❧✐❞❛❞ ❡❧ q✉❡ ❤❛ ❝♦♥s❡✲❣✉✐❞♦ ♠♦❞✐✜❝❛r ❞❡✜♥✐t✐✈❛♠❡♥t❡ ❧❛ ✐♠❛❣❡♥ q✉❡ ❧♦s ♠❛t❡♠át✐❝♦s t❡♥í❛♥ ❞❡❧ ❈á❧❝✉❧♦ ❞❡Pr♦❜❛❜✐❧✐❞❛❞❡s❀ ❧❛s ❝♦♥s❡❝✉❡♥❝✐❛s ❞❡ ❡s❡ ❤❡❝❤♦ s♦♥ ❡✈✐❞❡♥t❡s✿ ✉♥ ♥ú♠❡r♦ ❝r❡❝✐❡♥t❡ ❞❡♠❛t❡♠át✐❝♦s ❤❛♥ ✐❞♦ s✉♠❛♥❞♦ s✉s ❡s❢✉❡r③♦s ❡♥ ❧❛ ❝♦♥str✉❝❝✐ó♥ ❞❡❧ ❈á❧❝✉❧♦ ❞❡ Pr♦✲❜❛❜✐❧✐❞❛❞❡s ❤❛st❛ ❡❧ ♣✉♥t♦ ❞❡ q✉❡✱ ❤♦② ❡♥ ❞í❛✱ ❡s ✉♥❛ ❞❡ ❧❛s r❛♠❛s ♠ás ✐♥t❡r❡s❛♥t❡s✱❢ért✐❧❡s ② ❡❧❡❣❛♥t❡s ❞❡ ❧❛ ♠❛t❡♠át✐❝❛✳✳✳✑ ❙♦♥ ♠✉❝❤❛s ❧❛s ♦❜r❛s s♦❜r❡ ▼❡❞✐❞❛ ② Pr♦❜❛❜✐❧✐❞❛❞ ❡♥ ❧❛s q✉❡ s❡ ♣✉❡❞❡♥ ❡♥❝♦♥tr❛r❧♦s r❡s✉❧t❛❞♦s q✉❡ ♣r❡s❡♥t❛♠♦s ❡♥ ❡st❡ ♠❛♥✉❛❧ ❡♥ ✈❡rs✐♦♥❡s✱ ❛ ♠❡♥✉❞♦✱ ♠ás ❣❡♥❡r❛❧❡s② ❝♦♥ t♦❞❛s s✉s ❞❡♠♦str❛❝✐♦♥❡s❀ ♣❡r♠ít❛s❡♥♦s ❝✐t❛r ❛q✉í✱ ♣♦r ❡❥❡♠♣❧♦✱ ❧❛s s✐❣✉✐❡♥t❡s r❡❢❡r❡♥❝✐❛s✱ ❛ ❧❛s q✉❡ ❡st❡ tr❛t❛❞♦ ❞❡❜❡ t❛♥t♦✿ ❆s❤ ✭✶✾✼✷✮✱ ❘❡❛❧ ❆♥❛❧②s✐s ❛♥❞ Pr♦❜❛❜✐❧✐t②✱ ❆❝❛❞❡♠✐❝ Pr❡ss ✭② s✉ s✉❝❡s♦r ✏❆s❤✱❘✳❇✳✱ ❉♦❧❡❛♥s✲❉❛❞❡✱ ❈✳❆✳ ✭✷✵✵✵✮✱ Pr♦❜❛❜✐❧✐t② ✫ ▼❡❛s✉r❡ ❚❤❡♦r②✱ ✷♥❞ ❡❞✱ ❆❝❛✲❞❡♠✐❝ Pr❡ss✑✮❇❛✉❡r✱ ❍✳ ✭✶✾✾✺✮✱ Pr♦❜❛❜✐❧✐t② ❚❤❡♦r②✱ ❞❡ ●r✉②t❡r ❙t✉❞✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✳❇✐❧❧✐♥❣s❧❡②✱ P✳ ✭✶✾✾✺✮✱ Pr♦❜❛❜✐❧✐t② ❛♥❞ ▼❡❛s✉r❡✱ ❏✳ ❲✐❧❡② ✫ ❙♦♥s✳❈♦❤♥✳ ❉✳▲✳ ✭✶✾✽✵✮✱ ▼❡❛s✉r❡ ❚❤❡♦r②✱ ❇✐r❦❤ä✉s❡r ❱❡r❧❛❣✳ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDADM A N U A LE S U EX ❮◆❉■❈❊ ●❊◆❊❘❆▲❮◆❉■❈❊ ●❊◆❊❘❆▲ a −∞ ✮✳❇❛❞❛❥♦③✱ ✈❡r❛♥♦ ❞❡ ✷✵✵✽ ⋆ ♣r❡❝❡❞✐❞♦s ❞❡❧ sí♠❜♦❧♦ ✭↑✮❀ s❡ ♣r❡t❡♥❞❡ ✐♥❞✐❝❛r ❝♦♥ ❡❧❧♦ q✉❡✱ ❞❡ ❛❧❣ú♥ ♠♦❞♦✱ s♦♥ ❝♦♥✲ t✐♥✉❛❝✐ó♥ ❞❡❧ ♣r♦❜❧❡♠❛ ❛♥t❡r✐♦r✳ ❖tr♦s✱ q✉❡ s♦♥ ✐♥t❡r❡s❛♥t❡s ❝♦♠♣❧❡♠❡♥t♦s t❡ór✐❝♦sq✉❡ ♣✉❡❞❡♥ ♦❜✈✐❛rs❡ ❡♥ ✉♥❛ ♣r✐♠❡r❛ ❧❡❝t✉r❛ ❞❡❧ ♠❛♥✉❛❧ ② q✉❡✱ ❛ ♠❡♥✉❞♦✱ ♣r❡s❡♥t❛♥✉♥ ♠❛②♦r ❣r❛❞♦ ❞❡ ❞✐✜❝✉❧t❛❞✱ ✈❛♥ ♣r❡❝❡❞✐❞♦s ❞❡❧ sí♠❜♦❧♦ ✭ ✳❈❛❞❛ ❝❛♣ít✉❧♦ ✜♥❛❧✐③❛ ❝♦♥ ✉♥❛ ❝♦❧❡❝❝✐ó♥ ❞❡ ♣r♦❜❧❡♠❛s✳ ❆❧❣✉♥♦s ❞❡ ❡❧❧♦s ✈✐❡♥❡♥ ∞ ∞ = 0✳ ◆♦ s❡ ❞❡✜♥❡♥ ∞ − ∞ ♥✐ = ❉❛❝✉♥❤❛✲❈❛st❡❧❧❡✱ ❉✳✱ ❉✉✢♦✱ ▼✳ ✭✶✾✽✷✮✱ Pr♦❜❛❜✐❧✐tés ❡t ❙t❛t✐st✐q✉❡s✱ ✶✳ Pr♦❜❧é✲♠❡s à t❡♠♣s ✜①❡✱ ▼❛ss♦♥✳◆♦❣❛❧❡s✱ ❆✳●✳ ✭✷✵✵✹✮✱ ❇✐♦❡st❛❞íst✐❝❛ ❇ás✐❝❛✱ ❅❜❡❝❡❞❛r✐♦✳◆♦❣❛❧❡s✱ ❆✳●✳ ✭✶✾✾✽✮✱ ❊st❛❞íst✐❝❛ ▼❛t❡♠át✐❝❛✱ ❙❡r✈✐❝✐♦ ❞❡ P✉❜❧✐❝❛❝✐♦♥❡s ❞❡ ❧❛❯♥✐✈❡rs✐❞❛❞ ❞❡ ❊①tr❡♠❛❞✉r❛✳ a ∞ =−∞ s✐ b < 0✱ 0 · ∞ = ∞ · 0 = 0✱ ♠ét✐❝❛ ✉s✉❛❧✿ ❉❛❞♦s a ∈ R ② b ∈ [−∞, +∞]✱ s❡ ❞❡✜♥❡ a + ∞ = ∞ + a = ∞✱ a− ∞ = −∞ + a = −∞✱ ∞ + ∞ = ∞✱ −∞ − ∞ = −∞✱ b · ∞ = ∞ · b = ∞ s✐ b > 0✱ R = [−∞, +∞] ❞❡♥♦t❛rá ❧❛ r❡❝t❛ r❡❛❧ ❛♠♣❧✐❛❞❛✱ ❡♥ ❧❛ q✉❡ ❛s✉♠✐r❡♠♦s ❧❛ ❛r✐t✲ ✜♥✐t♦ ♦ ❡①✐st❡ ✉♥❛ ❛♣❧✐❝❛❝✐ó♥ ❜✐②❡❝t✐✈❛ ❞❡ N s♦❜r❡ é❧✳¯ N⊂ Z ⊂ Q ⊂ R ⊂ C ❞❡♥♦t❛rá♥ ❧♦s ❝♦♥❥✉♥t♦s ❞❡ ❧♦s ♥ú♠❡r♦s ♥❛t✉r❛❧❡s✱ ❡♥t❡r♦s✱ r❛❝✐♦♥❛❧❡s✱ r❡❛❧❡s ② ❝♦♠♣❧❡❥♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯♥ ❝♦♥❥✉♥t♦ s❡ ❞✐❝❡ ♥✉♠❡r❛❜❧❡ s✐ ❡s ❙❡ ❛s✉♠❡ ✉♥ ❜❛❣❛❥❡ ♠❛t❡♠át✐❝♦ ❜ás✐❝♦ ❞❡ t❡♦rí❛ ❞❡ ❝♦♥❥✉♥t♦s✱ á❧❣❡❜r❛ ❧✐♥❡❛❧✱❛♥á❧✐s✐s ♠❛t❡♠át✐❝♦ ❡♥ ✉♥❛ ② ✈❛r✐❛s ✈❛r✐❛❜❧❡s ② t♦♣♦❧♦❣í❛✳ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDADM A N U A LE S U EX ✶✵❮◆❉■❈❊ ●❊◆❊❘❆▲ ❊s♣❛❝✐♦s ▼❡❞✐❜❧❡s ❊♥ ❡st❛ s❡❝❝✐ó♥ s❡ ❞❡✜♥❡♥ ❧❛s ♥♦❝✐♦♥❡s ❞❡ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ ② ❞❡ ♠❡❞✐❞❛✳ Pr❡s❡♥✲ t❛♠♦s t❛♠❜✐é♥ ❛❧❣✉♥♦s ❞❡ ❧♦s r❡s✉❧t❛❞♦s ❜ás✐❝♦s q✉❡ ❤❛❜rá♥ ❞❡ s❡r út✐❧❡s ❡♥ ❡❧ r❡st♦❞❡ ❡st❡ ❝❛♣ít✉❧♦✳ ❊♥ ❧♦ q✉❡ s✐❣✉❡ Ω s❡rá ✉♥ ❝♦♥❥✉♥t♦ ♥♦ ✈❛❝í♦✳ ❊❋■◆■❈■Ó◆❉ ✶✳✶✳ ❙❡❛ A ✉♥❛ ❢❛♠✐❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✳ ❉✐r❡♠♦s q✉❡ A ❡s ✉♥ á❧❣❡❜r❛ ✭r❡s♣✳✱ σ✲á❧❣❡❜r❛✮ s✐ ❝♦♥t✐❡♥❡ ❛ Ω ② ❡s ❝❡rr❛❞♦ ♣♦r ❝♦♠♣❧❡♠❡♥t❛❝✐ó♥ ✭❡s ❞❡❝✐r✱ c s✐ A ❡stá ❡♥ A✱ s✉ ❝♦♠♣❧❡♠❡♥t❛r✐♦ A t❛♠❜✐é♥ ❡stá ❡♥ A✮ ② ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s ✜♥✐t❛s✭r❡s♣✳✱ ♥✉♠❡r❛❜❧❡s✮✳ ❙✐ A ❡s ✉♥❛ σ✲á❧❣❡❜r❛ ❞❡ ♣❛rt❡s ❞❡ Ω✱ ❡❧ ♣❛r (Ω, A) s❡ ❧❧❛♠❛rá✉♥ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡✱ ❧♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω q✉❡ ♣❡rt❡♥❡❝❡♥ ❛ A s❡ ❧❧❛♠❛rá♥ ❝♦♥❥✉♥t♦s♠❡❞✐❜❧❡s ✭A✲♠❡❞✐❜❧❡s✮ ♦ s✉❝❡s♦s✳ ❖❜s❡r✈❛❝✐ó♥ ✶✳✶✳ ▲♦♥❣✐t✉❞❡s✱ ár❡❛s✱ ✈♦❧ú♠❡♥❡s ② ♣r♦❜❛❜✐❧✐❞❛❞❡s s♦♥ ❡❥❡♠♣❧♦s ❝♦♥✲ ❝r❡t♦s ❞❡ ♠❡❞✐❞❛s ❡♥ ❡❧ s❡♥t✐❞♦ ❛♠♣❧✐♦ q✉❡ ❡st✉❞✐❛r❡♠♦s ❡♥ ❡st❡ ❝❛♣ít✉❧♦✳ ❯♥❛ ♠❡❞✐❞❛µ ❛s✐❣♥❛rá ❛ ❝✐❡rt♦s s✉❜❝♦♥❥✉♥t♦s A ❞❡ Ω ✉♥ ♥ú♠❡r♦ µ(A) q✉❡ ❧❧❛♠❛r❡♠♦s ♠❡❞✐❞❛ ❞❡A ✳ P♦r r❛③♦♥❡s q✉❡ ✈❡r❡♠♦s ♣♦st❡r✐♦r♠❡♥t❡✱ ♥♦ s✐❡♠♣r❡ ❡s ❛♣r♦♣✐❛❞♦ ❧❧❛♠❛r s✉❝❡s♦ ❛❝✉❛❧q✉✐❡r s✉❜❝♦♥❥✉♥t♦ ❞❡ Ω❀ ❡①✐❣✐r❡♠♦s ❡♥ ❝✉❛❧q✉✐❡r ❝❛s♦ q✉❡ ❧❛ ❝❧❛s❡ ❞❡ s✉❜❝♦♥❥✉♥✲ EX t♦s ❞❡ Ω q✉❡ s❡❛♥ ❞❡❝❧❛r❛❞♦s ♠❡❞✐❜❧❡s s❛t✐s❢❛❣❛ ❝✐❡rt❛s ♣r♦♣✐❡❞❛❞❡s✿ ❝♦♥❝r❡t❛♠❡♥t❡✱ U S ❡①✐❣✐r❡♠♦s q✉❡ s❡❛ ✉♥❛ σ✲á❧❣❡❜r❛✳ ❈♦♥s✐❞❡r❛❝✐♦♥❡s ♣r♦❜❛❜✐❧íst✐❝❛s ♣r♦♣♦r❝✐♦♥❛♥ ✉♥❛ LE ❜✉❡♥❛ ❥✉st✐✜❝❛❝✐ó♥ ❞❡ ❡s❛ ❛✜r♠❛❝✐ó♥✳ ❙✉♣♦♥❣❛♠♦s q✉❡ Ω ❡s ❡❧ ❝♦♥❥✉♥t♦ ❞❡ r❡s✉❧t❛✲ A U ❞♦s ♣♦s✐❜❧❡s ❞❡ ✉♥ ❝✐❡rt♦ ❡①♣❡r✐♠❡♥t♦ ❛❧❡❛t♦r✐♦❀ ❛ ❝✐❡rt♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω s❡ ❧❡s N A ❧❧❛♠❛rá s✉❝❡s♦s ② s❡ ❧❡s ❛s✐❣♥❛rá ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞✳ ■♥t✉✐t✐✈❛♠❡♥t❡✱ s✐ A ❡s ✉♥ s✉❝❡s♦✱ M ❞✐r❡♠♦s q✉❡ ❡❧ s✉❝❡s♦ A ❤❛ ♦❝✉rr✐❞♦ ❡♥ ✉♥❛ r❡❛❧✐③❛❝✐ó♥ ❞❡❧ ❡①♣❡r✐♠❡♥t♦ s✐ ❡❧ r❡s✉❧t❛❞♦ ✶✷ ❈❆P❮❚❯▲❖ ✶✳ ❊❙P❆❈■❖❙ ▼❊❉■❇▲❊❙ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ω♦❜t❡♥✐❞♦ ❡♥ ❡s❛ r❡❛❧✐③❛❝✐ó♥ ❡stá ❡♥ A✳ ❙❡❛ ❝✉❛❧ s❡❛ ❡❧ r❡s✉❧t❛❞♦ ω ❞❡❧ ❡①♣❡r✐♠❡♥t♦✱ Ω♦❝✉rr❡❀ ♣♦r t❛♥t♦✱ Ω ❞❡❜❡ s❡r ✉♥ s✉❝❡s♦❀ ❧❡ ❧❧❛♠❛r❡♠♦s s✉❝❡s♦ s❡❣✉r♦✳ ❙✐ ♣❡♥s❛r ❡♥ ❧❛ ♦❝✉rr❡♥❝✐❛ ❞❡ A t✐❡♥❡ ♣❡r❢❡❝t♦ s❡♥t✐❞♦ ♣♦r s❡r A ✉♥ s✉❝❡s♦✱ t❛♠❜✐é♥ t✐❡♥❡ ♣❡r❢❡❝t♦ c s❡♥t✐❞♦ ♣❡♥s❛r ❡♥ ❧❛ ♥♦ ♦❝✉rr❡♥❝✐❛ ❞❡ A✱ q✉❡ ❡q✉✐✈❛❧❡ ❛ ❧❛ ♦❝✉rr❡♥❝✐❛ ❞❡ A ✭❝♦♠✲ c ♣❧❡♠❡♥t❛r✐♦ ❞❡ A✮❀ ❡s✱ ♣♦r t❛♥t♦✱ r❛③♦♥❛❜❧❡ ❡①✐❣✐r q✉❡ A s❡❛ ✉♥ s✉❝❡s♦ s✐ A ❧♦ ❡s✳ ❙✐ t✐❡♥❡ s❡♥t✐❞♦ ♣❡♥s❛r ❡♥ ❧❛ ♦❝✉rr❡♥❝✐❛ ❞❡ A ⊂ Ω ② ❡♥ ❧❛ ♦❝✉rr❡♥❝✐❛ ❞❡ B ⊂ Ω ♣♦r s❡rA ② B s✉❝❡s♦s✱ t❛♠❜✐é♥ t✐❡♥❡ s❡♥t✐❞♦ ♣❡♥s❛r ❡♥ ❧❛ ♦❝✉rr❡♥❝✐❛ ❞❡ A ∪ B ✭s✐ s❛❜❡♠♦s❞❡❝✐❞✐r ❝✉á♥❞♦ ♦❝✉rr❡ A ② ❝✉á♥❞♦ ♦❝✉rr❡ B✱ t❛♠❜✐é♥ s❛❜r❡♠♦s ❞❡❝✐❞✐r ❝✉❛♥❞♦ ♦❝✉rr❡A ♦ B✮❀ ❡①✐❣✐r❡♠♦s ❡♥t♦♥❝❡s q✉❡ ❧❛ ✉♥✐ó♥ ✜♥✐t❛ ❞❡ s✉❝❡s♦s s❡❛ ✉♥ s✉❝❡s♦✳ ▼❡♥♦s✐♥t✉✐t✐✈❛ ❡s ❧❛ ❡①✐❣❡♥❝✐❛ ❞❡ ❡st❛❜✐❧✐❞❛❞ ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s ♥✉♠❡r❛❜❧❡s❀ ❧❛ r❡s♣✉❡st❛ ♠ás❝♦♥✈✐♥❝❡♥t❡ ❡s q✉❡✱ ❞❡ ❡s❡ ♠♦❞♦✱ s❡ ♦❜t✐❡♥❡ ✉♥❛ t❡♦rí❛ ♠❛t❡♠át✐❝❛ ♠ás r✐❝❛✳ ❖❜s❡r✈❛❝✐ó♥ ✶✳✷✳ ▲♦s ♦rí❣❡♥❡s ② ♣r✐♠❡r♦s ♣❛s♦s ❞❡ ❧❛s t❡♦rí❛s ❞❡ ❧❛ ♠❡❞✐❞❛ ② ❞❡ ❧❛ ♣r♦❜❛❜✐❧✐❞❛❞ s♦♥ ❞✐❢❡r❡♥t❡s✳ ❋✉❡ ❑♦❧♠♦❣♦r♦✈ q✉✐❡♥ ♣r♦♣✉s♦ ♣♦r ♣r✐♠❡r❛ ✈❡③ ✉♥♣✉♥t♦ ❞❡ ♣❛rt✐❞❛ ❢♦r♠❛❧ ❝♦♠ú♥ ♣❛r❛ ❛♠❜❛s t❡♦rí❛s ✭q✉❡ ❝♦♥s❡r✈❛♥ ❛ú♥ s✉s ♣r♦♣✐♦s♦❜❥❡t✐✈♦s✮✳ ❉❡❜✐❞♦ ❛ ❡❧❧♦✱ s✉❜s✐st❡ ✉♥❛ ❞♦❜❧❡ t❡r♠✐♥♦❧♦❣í❛ ♣❛r❛ ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s✐❞é♥t✐❝♦s q✉❡ ❝♦♠♣r♦❜❛r❡♠♦s ❡♥ r❡♣❡t✐❞❛s ♦❝❛s✐♦♥❡s ❛ ❧♦ ❧❛r❣♦ ❞❡ ❡st❡ ♠❛♥✉❛❧ ② q✉❡✱ ❞❡ ❤❡❝❤♦✱ ②❛ ❤❡♠♦s ❝♦♠♣r♦❜❛❞♦✿ s✐ ❡♥ t❡♦rí❛ ❞❡ ❧❛ ♠❡❞✐❞❛ ❧❧❛♠❛♠♦s ❝♦♥❥✉♥t♦s♠❡❞✐❜❧❡s ❛ ❧♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♥❛ σ✲á❧❣❡❜r❛✱ ❡♥ ♣r♦❜❛❜✐❧✐❞❛❞ ❧❡s ❧❧❛♠❛♠♦s s✉❝❡s♦s✳ Pr♦♣♦s✐❝✐ó♥ ✶✳✶✳ ✭❛✮ ❙✐ A ❡s ✉♥ á❧❣❡❜r❛ ✭r❡s♣✳✱ σ✲á❧❣❡❜r❛✮ ❞❡ ♣❛rt❡s ❞❡ Ω✱ A ❡s ❝❡rr❛❞❛ ❢r❡♥t❡ ❛ ❧❛ ❞✐❢❡r❡♥❝✐❛ ❞❡ ❝♦♥❥✉♥t♦s ❡ ✐♥t❡rs❡❝❝✐♦♥❡s ✜♥✐t❛s ✭r❡s♣✳✱ ♥✉♠❡r❛❜❧❡s✮② ❝♦♥t✐❡♥❡ ❛❞❡♠ás ❛❧ ❝♦♥❥✉♥t♦ ✈❛❝í♦✳ ✭❜✮ ❙✐ A ❡s ✉♥ á❧❣❡❜r❛ ❞❡ ♣❛rt❡s ❞❡ Ω✱ s❡rá t❛♠❜✐é♥ ✉♥❛ σ✲á❧❣❡❜r❛ s✐ ❛❞❡♠ás ❡s❝❡rr❛❞❛ ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s ♥✉♠❡r❛❜❧❡s ❝r❡❝✐❡♥t❡s ♦ ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s ♥✉♠❡r❛❜❧❡s ❞✐s❥✉♥✲ t❛s✳❉❡♠♦str❛❝✐ó♥✳ ✭❛✮ ❚r✐✈✐❛❧✳ )A = B = C n nn n n n n n ✭❜✮ ❙✐ (A ❡s ✉♥❛ s✉❝❡s✐ó♥ ❡♥ A✱ ❡♥t♦♥❝❡s A := ∪ ∪ ∪ ❞♦♥❞❡ n n−1 B = A = A A, C n i ② C n n i ✳ ❙✐❡♥❞♦ A ✉♥ á❧❣❡❜r❛✱ s❡ ✈❡r✐✜❝❛ q✉❡ B n n ∪ i=1 \ ∪ i=1∈ A✱ ♣❛r❛ ❝❛❞❛ n✳ ❙✐ A s❡ s✉♣♦♥❡ ❡st❛❜❧❡ ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s ♥✉♠❡r❛❜❧❡s ❝r❡❝✐❡♥t❡s ✭r❡s♣✳✱ EX B C n n n n ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s ♥✉♠❡r❛❜❧❡s ❞✐s❥✉♥t❛s✮ ❡♥t♦♥❝❡s A = ∪ ∈ A ✭r❡s♣✳✱ A = ∪ ∈ U S A✮✳ LE A U N ❆♥t❡s ❞❡ ✈❡r ❡❥❡♠♣❧♦s ❞❡ σ✲á❧❣❡❜r❛s✱ ♣r♦❜❛r❡♠♦s ✉♥❛ ♣r♦♣♦s✐❝✐ó♥ q✉❡ ♣r♦♣♦r❝✐♦♥❛ A M ✉♥ ♠ét♦❞♦ ✭♣♦❝♦ ❞❡s❝r✐♣t✐✈♦✱ ♣❡r♦ ❢r❡❝✉❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞♦✮ ♣❛r❛ ❝♦♥str✉✐r❧❛s✳ ✶✸ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDADPr♦♣♦s✐❝✐ó♥ ✶✳✷✳ ▲❛ ✐♥t❡rs❡❝❝✐ó♥ ❛r❜✐tr❛r✐❛ ❞❡ σ✲á❧❣❡❜r❛s ❞❡ ♣❛rt❡s ❞❡ ✉♥ ❝♦♥❥✉♥t♦ Ω❡s ✉♥❛ σ✲á❧❣❡❜r❛ ❡♥ Ω✳ ❆sí✱ s✐ C ❡s ✉♥❛ ❢❛♠✐❧✐❛ ❞❡ ♣❛rt❡s ❞❡ Ω✱ ❡①✐st❡ ❧❛ ♠ás ♣❡q✉❡ñ❛ σ✲á❧❣❡❜r❛ ❡♥ Ω q✉❡ ❝♦♥t✐❡♥❡ ❛ C❀ ❧❛ ❧❧❛♠❛r❡♠♦s σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r C② ❧❛ ❞❡♥♦t❛r❡♠♦s σ(C)✳❉❡♠♦str❛❝✐ó♥✳ ▲❛ ♣r✐♠❡r❛ ❛✜r♠❛❝✐ó♥ ❡s tr✐✈✐❛❧✳ ▲❛ s❡❣✉♥❞❛ ❡s ❝♦♥s❡❝✉❡♥❝✐❛ ❞❡ ❧❛♣r✐♠❡r❛✱ ♣✉❡s σ(C) ❡s ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡ t♦❞❛s ❧❛s σ✲á❧❣❡❜r❛s q✉❡ ❝♦♥t✐❡♥❡♥ ❛ C✱❝♦♥❥✉♥t♦ ❞❡ σ✲á❧❣❡❜r❛s q✉❡ ❡s ♥♦ ✈❛❝í♦ ♣✉❡s ❝♦♥t✐❡♥❡ ❛ ❧❛ σ✲á❧❣❡❜r❛ P(Ω) ❞❡ t♦❞♦s❧♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✳ ❊❥❡♠♣❧♦ ✶✳✶✳ ✭σ✲á❧❣❡❜r❛ ❞✐s❝r❡t❛✮ ❊❧ ❝♦♥❥✉♥t♦ P(Ω) ❞❡ t♦❞♦s ❧♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ❡s tr✐✈✐❛❧♠❡♥t❡ ✉♥❛ σ✲á❧❣❡❜r❛ ❡♥ Ω q✉❡ ❧❧❛♠❛r❡♠♦s ❞✐s❝r❡t❛✳ ◆♦ ♦❜st❛♥t❡✱ ❧❧❛♠❛r❡♠♦s❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ ❞✐s❝r❡t♦ ✉♥ ❝♦♥❥✉♥t♦ ♥✉♠❡r❛❜❧❡ ♣r♦✈✐st♦ ❞❡ ❧❛ σ✲á❧❣❡❜r❛ ❞❡ t♦❞❛s s✉s♣❛rt❡s✳ ❊♥ ❧♦ q✉❡ s✐❣✉❡✱ ❝✉❛❧q✉✐❡r ❝♦♥❥✉♥t♦ ♥✉♠❡r❛❜❧❡ s❡ s✉♣♦♥❞rá ♣r♦✈✐st♦ ❞❡ ❧❛σ ✲á❧❣❡❜r❛ ❞✐s❝r❡t❛✱ ❛ ♠❡♥♦s q✉❡✱ ❡①♣r❡s❛♠❡♥t❡✱ s❡ ✐♥❞✐q✉❡ ❧♦ ❝♦♥tr❛r✐♦✳ ❊❥❡♠♣❧♦ ✶✳✷✳ ✭σ✲á❧❣❡❜r❛ tr✐✈✐❛❧ ♦ ❣r♦s❡r❛✮ {∅, Ω} ❡s ❧❛ ♠ás ♣❡q✉❡ñ❛ σ✲á❧❣❡❜r❛ q✉❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❡♥ Ω✳ ❙❡ ❧❧❛♠❛rá σ✲á❧❣❡❜r❛ ❣r♦s❡r❛ ♦ σ✲á❧❣❡❜r❛ tr✐✈✐❛❧✳ ❊❥❡♠♣❧♦ ✶✳✸✳ ❙✐ A ❡s ✉♥ s✉❜❝♦♥❥✉♥t♦ ❞❡ Ω✱ ❧❛ ♠ás ♣❡q✉❡ñ❛ σ✲á❧❣❡❜r❛ q✉❡ ❝♦♥t✐❡♥❡c , Ω , . . . , A 1 n ❛ A ❡s {∅, A, A }✳ ❙✐ A ❡s ✉♥❛ ♣❛rt✐❝✐ó♥ ✜♥✐t❛ ❞❡ Ω ✭❡s ❞❡❝✐r✱ s✉❜❝♦♥❥✉♥t♦s❞♦s ❛ ❞♦s ❞✐s❥✉♥t♦s q✉❡ r❡❝✉❜r❡♥ Ω✮✱ ❡♥t♦♥❝❡s ❧❛ ♠ás ♣❡q✉❡ñ❛ σ✲á❧❣❡❜r❛ q✉❡ ❝♦♥t✐❡♥❡❛ ❧♦s A i ❡stá ❢♦r♠❛❞❛ ♣♦r ❡❧ ❝♦♥❥✉♥t♦ ✈❛❝í♦ ② ❧❛s ✉♥✐♦♥❡s ✜♥✐t❛s ❞❡ A i ✬s✳ n❊❥❡♠♣❧♦ ✶✳✹✳ ✭σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧✮ ▲❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ R ✱ q✉❡ ❞❡♥♦t❛r❡♠♦sn ♣♦r R ✭♦✱ s✐♠♣❧❡♠❡♥t❡✱ R s✐ n = 1✮✱ s❡ ❞❡✜♥❡ ❝♦♠♦ ❧❛ σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r ❧♦s n : x < a i ✐♥t❡r✈❛❧♦s n✲❞✐♠❡♥s✐♦♥❛❧❡s ❞❡ ❧❛ ❢♦r♠❛ {x ∈ R } ♣❛r❛ ❛❧❣ú♥ a ∈ R ② ❛❧❣ú♥ n 1≤ i ≤ n✳ ▲❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ R ❡s t❛♠❜✐é♥ ❧❛ σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r ❝❛❞❛ n ✉♥❛ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢❛♠✐❧✐❛s ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ R ✿ ❧❛ ❢❛♠✐❧✐❛ ❞❡ ❧♦s ❛❜✐❡rt♦s ❞❡ n R✱ ❧❛ ❞❡ ❧♦s ❝♦♠♣❛❝t♦s✱ ❧❛ ❞❡ ❧♦s ✐♥t❡r✈❛❧♦s n✲❞✐♠❡♥s✐♦♥❛❧❡s ❝❡rr❛❞♦s ② ❛❝♦t❛❞♦s ✭✈❡r n Pr♦❜❧❡♠❛ ✶✳✽ ✮✳ ❊♥ ❧♦ q✉❡ s✐❣✉❡✱ R s❡ s✉♣♦♥❞rá ♣r♦✈✐st♦ ❞❡ ❧❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧✱ ❛♠❡♥♦s q✉❡✱ ❡①♣r❡s❛♠❡♥t❡✱ s❡ ✐♥❞✐q✉❡ ❧♦ ❝♦♥tr❛r✐♦✳ R ❊❥❡♠♣❧♦ ✶✳✺✳ ¯R ❞❡♥♦t❛rá ❧❛ r❡❝t❛ r❡❛❧ ❛♠♣❧✐❛❞❛ ② ¯ R ❧❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ ¯ q✉❡ s❡ ❞❡✜♥❡ ❝♦♠♦ ❧❛ σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r ❧♦s ❝♦♥❥✉♥t♦s ❞❡ ❧❛ ❢♦r♠❛ [−∞, x]✱ n n nEX x R∈ ¯ ✳ ❆♥á❧♦❣❛♠❡♥t❡ s❡ ❞❡✜♥❡ ❧❛ σ✲á❧❣❡❜r❛ ¯ R ❞❡ ❇♦r❡❧ ❡♥ ¯R ✳ ❊♥ ❧♦ q✉❡ s✐❣✉❡✱ ¯R U s❡ s✉♣♦♥❞rá ♣r♦✈✐st♦ ❞❡ ❧❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧✱ ❛ ♠❡♥♦s q✉❡✱ ❡①♣r❡s❛♠❡♥t❡✱ s❡ ✐♥❞✐q✉❡ S LE ❧♦ ❝♦♥tr❛r✐♦✳ A U❊❥❡♠♣❧♦ ✶✳✻✳ ❊♥ ❣❡♥❡r❛❧✱ ❧❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❡♥ ✉♥ ❡s♣❛❝✐♦ t♦♣♦❧ó❣✐❝♦ ❡s ❧❛ ❡♥✲ N A ❣❡♥❞r❛❞❛ ♣♦r s✉s ❛❜✐❡rt♦s❀ ❧♦s s✉❝❡s♦s ❞❡ ❡st❛ σ✲á❧❣❡❜r❛ s❡ ❧❧❛♠❛♥ ❝♦♥❥✉♥t♦s ❞❡ ❇♦r❡❧ M ♦ ❜♦r❡❧✐❛♥♦s✳ ✶✹ ❈❆P❮❚❯▲❖ ✶✳ ❊❙P❆❈■❖❙ ▼❊❉■❇▲❊❙ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDADi , i ) ❊❥❡♠♣❧♦ ✶✳✼✳ ✭σ✲á❧❣❡❜r❛ ♣r♦❞✉❝t♦✮ ❙❡❛♥ (Ω A ✱ i = 1, 2✱ ❡s♣❛❝✐♦s ♠❡❞✐❜❧❡s ② ❤❛❣❛✲ 1 2 1 2 ♠♦s Ω = Ω ×Ω ✳ ❯♥ r❡❝tá♥❣✉❧♦ ♠❡❞✐❜❧❡ ❡♥ Ω ❡s ✉♥ ❝♦♥❥✉♥t♦ ❞❡ ❧❛ ❢♦r♠❛ A ×A ❝♦♥A i i ∈ A ✱ i = 1, 2✳ ▲❛ ❢❛♠✐❧✐❛ ❞❡ ❧♦s r❡❝tá♥❣✉❧♦s ♠❡❞✐❜❧❡s ♥♦ ❡s ✉♥❛ σ✲á❧❣❡❜r❛✱ s❛❧✈♦❡♥ ❝❛s♦s tr✐✈✐❛❧❡s ✭✈❡r Pr♦❜❧❡♠❛ ✶✳✻ ✮❀ ❧❛ σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r ❡❧❧♦s s❡ ❧❧❛♠❛ráσ 1 2 ✲á❧❣❡❜r❛ ♣r♦❞✉❝t♦ ② s❡ ❞❡♥♦t❛rá A ×A ✳ ❆♥á❧♦❣❛♠❡♥t❡ s❡ ❞❡✜♥❡ ❡❧ ♣r♦❞✉❝t♦ ❞❡ ✉♥❛, ) ❝❛♥t✐❞❛❞ ✜♥✐t❛ ❞❡ ❡s♣❛❝✐♦s ♠❡❞✐❜❧❡s✳ ❙✐ (Ω i i ❡s ✉♥❛ ❢❛♠✐❧✐❛ ❞❡ ❡s♣❛❝✐♦s ♠❡❞✐❜❧❡sA i∈I Ω ii ② Ω = Q ✱ ❧❧❛♠❛r❡♠♦s σ✲á❧❣❡❜r❛ ♣r♦❞✉❝t♦ ❡♥ Ω✱ ② ❧❛ ❞❡♥♦t❛r❡♠♦s Q A ✱ ❧❛ i∈Ii∈I σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r ❧♦s r❡❝tá♥❣✉❧♦s ♠❡❞✐❜❧❡s ✭❞❡❢✳✿ ✉♥ r❡❝tá♥❣✉❧♦ ♠❡❞✐❜❧❡ ❡♥ Ω A = Ω i i i i i ❡s ✉♥ ♣r♦❞✉❝t♦ ❞❡ ❧❛ ❢♦r♠❛ Q ✱ ❞♦♥❞❡ A ∈ A ✱ ∀i ② A s✐ i ∈ I \ I i∈I n ♣❛r❛ ✉♥ ❝✐❡rt♦ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ I ❞❡ I✮✳ ▲❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❡♥ R ❝♦✐♥❝✐❞❡ ❝♦♥ n ❡❧ ♣r♦❞✉❝t♦ ❞❡ n ❝♦♣✐❛s ❞❡ ❧❛ ❞❡ ❇♦r❡❧ ❡♥ R❀ ❧❛ ♥♦t❛❝✐ó♥ R ✉t✐❧✐③❛❞❛ ♣❛r❛ ❡❧❧❛ ♥♦♣✉❡❞❡ ❞❛r ❧✉❣❛r ❛ ❝♦♥❢✉s✐ó♥✳ ❊❥❡♠♣❧♦ ✶✳✽✳ ✭σ✲á❧❣❡❜r❛ ✐♥❞✉❝✐❞❛ ❡♥ ✉♥ s✉❜❡s♣❛❝✐♦✮ ❙✐ (Ω, A) ❡s ✉♥ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ := B ② B ❡s ✉♥ s✉❜❝♦♥❥✉♥t♦ ♥♦ ✈❛❝í♦ ❞❡ Ω ❡♥t♦♥❝❡s A {A ∩ B : A ∈ A} ❡s ✉♥❛ σ✲ B á❧❣❡❜r❛ ❡♥ B✳ ❙✐ B ∈ A ❧♦s ❡❧❡♠❡♥t♦s ❞❡ A s♦♥ A✲♠❡❞✐❜❧❡s✳ ❉❡ ❢♦r♠❛ ❣❡♥❡r❛❧✱ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ B ❞❡ ✉♥❛ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ (Ω, A) s❡ s✉♣♦♥❞rá ♣r♦✈✐st♦✱ ❛ ♠❡♥♦s q✉❡ s❡✐♥❞✐q✉❡ ❧♦ ❝♦♥tr❛r✐♦✱ ❞❡ ❧❛ σ✲á❧❣❡❜r❛ A B ✭✈❡r Pr♦❜❧❡♠❛ ✶✳✾ ✮✳ ❊❥❡♠♣❧♦ ✶✳✾✳ ✭❙✉❜✲σ✲á❧❣❡❜r❛✮ ❙✐ (Ω, A) ❡s ✉♥ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ ② B ❡s ✉♥❛ σ✲á❧❣❡❜r❛ ❡♥ Ω t❛❧ q✉❡ B ⊂ A✱ ❞✐r❡♠♦s q✉❡ B ❡s ✉♥❛ s✉❜✲σ✲á❧❣❡❜r❛ ❞❡ A✳ P♦r ❡❥❡♠♣❧♦✱ c , R{∅, [0, 1], [0, 1] } ❡s ✉♥❛ s✉❜✲σ✲á❧❣❡❜r❛ ❞❡ R✱ ② ést❛ ❧♦ ❡s ❞❡ P(R) ✭✈❡r Pr♦❜❧❡✲♠❛ ✶✵✳✷✻ ✮✳ ❊❧ t❡♦r❡♠❛ s✐❣✉✐❡♥t❡ ❡s ✉♥ r❡s✉❧t❛❞♦ ❞❡ ❉②♥❦✐♥ q✉❡ s❡ ✉t✐❧✐③❛ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❡❧ ✶ tr❛❜❛❥♦ ❞❡ ♣r♦❜❛r s✐ ✉♥❛ ❝✐❡rt❛ ❢❛♠✐❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ❡s ✉♥❛ σ✲á❧❣❡❜r❛✳✭ ✮ ❊❋■◆■❈■Ó◆❉ ✶✳✷✳ ❙❡❛ C ✉♥❛ ❢❛♠✐❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω✳ ✭❛✮ ❉✐r❡♠♦s q✉❡ C ❡s ✉♥ π✲s✐st❡♠❛ s✐ ❡s ❡st❛❜❧❡ ❢r❡♥t❡ ❛ ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ✜♥✐t❛✳✭❜✮ ❉✐r❡♠♦s q✉❡ C ❡s ✉♥ ❞✲s✐st❡♠❛ ✭♦ ✉♥❛ ❝❧❛s❡ ❞❡ ❉②♥❦✐♥✮ s✐ Ω ∈ C✱ ❡s ❡st❛❜❧❡ ❢r❡♥t❡ ❛ ❧❛ ✉♥✐ó♥ ♥✉♠❡r❛❜❧❡ ❝r❡❝✐❡♥t❡ ② ❢r❡♥t❡ ❛ ❞✐❢❡r❡♥❝✐❛s ♣r♦♣✐❛s ✭❡st♦ ú❧t✐♠♦ q✉✐❡r❡❞❡❝✐r q✉❡ s❡ ✈❡r✐✜❝❛ ❧❛ ✐♠♣❧✐❝❛❝✐ó♥ [A, B ∈ C, A ⊂ B] =⇒ [B \ A ∈ C]✮✳ EX ■❣✉❛❧ q✉❡ ♦❝✉rrí❛ ❝♦♥ σ✲á❧❣❡❜r❛s✱ ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡ ✉♥❛ ❝♦❧❡❝❝✐ó♥ ❛r❜✐tr❛r✐❛ ❞❡ ❞✲ U S s✐st❡♠❛s ❡s ✉♥ ❞✲s✐st❡♠❛ ② ❧❛ ❢❛♠✐❧✐❛ ❞❡ t♦❞♦s ❧♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ❡s ✉♥ ❞✲s✐st❡♠❛❀ LE A✶ ❈♦♥ ❡s❡ ♠✐s♠♦ ♦❜❥❡t✐✈♦ s❡ s✉❡❧❡ ✉s❛r ❡♥ ❧❛ ❧✐t❡r❛t✉r❛ ❡❧ ❧❧❛♠❛❞♦ t❡♦r❡♠❛ ❞❡ ❧❛ ❝❧❛s❡ ♠♦♥ót♦♥❛❀U NA ✈é❛s❡✱ ♣♦r ❡❥❡♠♣❧♦✱ ❆s❤ ✭✶✾✼✷✮✳ ◆♦ ♦❜st❛♥t❡✱ s✉❡❧❡ s❡r ♠ás s❡♥❝✐❧❧♦ ❞❡ ♠❛♥❡❥❛r ❡❧ r❡s✉❧t❛❞♦ ❞❡ M ❉②♥❦✐♥✳ ✶✺ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ♣♦r t❛♥t♦✱ ❞❛❞❛ ✉♥❛ ❢❛♠✐❧✐❛ C ❞❡ ♣❛rt❡s ❞❡ Ω✱ ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡ t♦❞♦s ❧♦s ❞✲s✐st❡♠❛s q✉❡ ❝♦♥t✐❡♥❡♥ ❛ C ❡s ✉♥ ❞✲s✐st❡♠❛ ② ❡s ❡❧ ♠ás ♣❡q✉❡ñ♦ ❞✲s✐st❡♠❛ q✉❡ ❝♦♥t✐❡♥❡ ❛ C❀s❡ ❧❧❛♠❛ ❞✲s✐st❡♠❛ ❡♥❣❡♥❞r❛❞♦ ♣♦r C✳ ❊❖❘❊▼❆❚ ✶✳✶✳ ✭❚❡♦r❡♠❛ ❞❡ ❉②♥❦✐♥✮ ❙✐ C ❡s ✉♥ π✲s✐st❡♠❛ ❡♥ Ω ② A ✉♥ ❞✲s✐st❡♠❛ q✉❡ ❝♦♥t✐❡♥❡ ❛ C✱ ❡♥t♦♥❝❡s A ❝♦♥t✐❡♥❡ ❛ ❧❛ σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r C✳ ❊♥ ♣❛rt✐❝✉❧❛r✱❡❧ ❞✲s✐st❡♠❛ ❡♥❣❡♥❞r❛❞♦ ♣♦r ✉♥ π✲s✐st❡♠❛ ❝♦✐♥❝✐❞❡ ❝♦♥ ❧❛ σ✲á❧❣❡❜r❛ ❡♥❣❡♥❞r❛❞❛ ♣♦r❡❧ π✲s✐st❡♠❛✳❉❡♠♦str❛❝✐ó♥✳ ❙❡❛ D ❡❧ ❞✲s✐st❡♠❛ ❡♥❣❡♥❞r❛❞♦ ♣♦r C✳ P✉❡st♦ q✉❡ ❧❛s σ✲á❧❣❡❜r❛s s♦♥ ❞✲ s✐st❡♠❛s✱ s❡ ✈❡r✐✜❝❛ q✉❡ D ⊂ σ(C)✳ P❛r❛ ♣r♦❜❛r ❧❛ ♦tr❛ ✐♥❝❧✉s✐ó♥✱ ❜❛st❛rá ♣r♦❜❛r q✉❡D ❡s ✉♥❛ σ✲á❧❣❡❜r❛✳ ❈♦♠❡♥③❛r❡♠♦s ♣r♦❜❛♥❞♦ q✉❡ D ❡s ❡st❛❜❧❡ ♣♦r ✐♥t❡rs❡❝❝✐ó♥ ✜♥✐t❛✳ := 1 ❈♦♥s✐❞❡r❡♠♦s ❧❛ ❢❛♠✐❧✐❛ D {A ∈ D : A ∩ C ∈ D ♣❛r❛ ❝❛❞❛ C ∈ C}✳ P✉❡st♦ q✉❡ C 1 1 1 ❡s ❡st❛❜❧❡ ♣♦r ✐♥t❡rs❡❝❝✐ó♥ ✜♥✐t❛✱ C ⊂ D ❀ ❛❞❡♠ás D ❡s ✉♥ ❞✲s✐st❡♠❛ ✭Ω ∈ D ♣✉❡sA ) (A n n n n C ⊂ D✱ ② ❧❛s ✐❣✉❛❧❞❛❞❡s (A \ B) ∩ C = (A ∩ C) \ (B ∩ C) ② (∪ ∩ C = ∪ ∩ C) 1 ✐♠♣❧✐❝❛♥ q✉❡ D ❡s ❝❡rr❛❞❛ ❢r❡♥t❡ ❛ ❧❛ ❢♦r♠❛❝✐ó♥ ❞❡ ❞✐❢❡r❡♥❝✐❛s ♣r♦♣✐❛s ② ❢r❡♥t❡ ❛ 1 1 ✉♥✐♦♥❡s ♥✉♠❡r❛❜❧❡s ❝r❡❝✐❡♥t❡s✮✳ ❊♥t♦♥❝❡s D ⊂ D ✳ P❡r♦ ♣♦r ❞❡✜♥✐❝✐ó♥ ❞❡ D s❡ t✐❡♥❡ 1 1 q✉❡ D ⊃ D ❀ ❧✉❡❣♦ D = D ✳:= 2 ❍❛❣❛♠♦s ❛❤♦r❛ D {B ∈ D : A ∩ B ∈ D ♣❛r❛ ❝❛❞❛ A ∈ D}✳ P✉❡st♦ q✉❡= 12 D D✱ s❡ t✐❡♥❡ q✉❡ C ⊂ D ✳ ❆r❣✉♠❡♥t♦s ❛♥á❧♦❣♦s ❛ ❧♦s ♣r❡❝❡❞❡♥t❡s ♣r✉❡❜❛♥ q✉❡2 2 = D ❡s ✉♥ ❞✲s✐st❡♠❛ ② q✉❡ D D✳ P♦r t❛♥t♦✱ D ❡s ❡st❛❜❧❡ ♣♦r ✐♥t❡rs❡❝❝✐ó♥ ✜♥✐t❛✳❉❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡ ❞✲s✐st❡♠❛ s❡ ❞❡❞✉❝❡ q✉❡ D ❡s ❡st❛❜❧❡ ♣♦r ♣❛s♦ ❛❧ ❝♦♠♣❧❡♠❡♥✲ c = Ω t❛r✐♦ ✭s✐ A ∈ D ❡♥t♦♥❝❡s A \ A ∈ D✮❀ ♣✉❡st♦ q✉❡ ❤❡♠♦s ✈✐st♦ q✉❡ ❡s ❡st❛❜❧❡♣♦r ✐♥t❡rs❡❝❝✐ó♥ ✜♥✐t❛✱ D ❡s ✉♥ á❧❣❡❜r❛✳ P❡r♦ ✉♥ á❧❣❡❜r❛ ❡st❛❜❧❡ ❢r❡♥t❡ ❛ ✉♥✐♦♥❡s❝r❡❝✐❡♥t❡s ❡s ✉♥❛ σ✲á❧❣❡❜r❛✳ EX U S LE A U N A M ✶✻ ❈❆P❮❚❯▲❖ ✶✳ ❊❙P❆❈■❖❙ ▼❊❉■❇▲❊❙ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✶c cPr♦❜❧❡♠❛ ✶✳✶✳ ❙✐ A ② B s♦♥ s✉❝❡s♦s ❞✐s❥✉♥t♦s✱ ❞❡t❡r♠✐♥❛r A \ B✱ A△B ② A ✳ ∩ B 1 1 1 1 , , 1 n nPr♦❜❧❡♠❛ ✶✳✷✳ ❛✮ ❈❛❧❝✉❧❛r ∩ − ② ∩ − ✳n n n n 1 = , 1 A := A n n n k ❜✮ P❛r❛ ❝❛❞❛ n ∈ N✱ s❡❛ A − ✳ ❉❡t❡r♠✐♥❛r l´ım sup n ∩ ∪ k≥n ② n l´ım inf A := A n n n k ∪ ∩ k≥n ✳ 1 1 = , 1❝✮ ❘❡♣❡t✐r ❡❧ ❛♣❛rt❛❞♦ ❛♥t❡r✐♦r ♣❛r❛ A n s✐ n ❡s ♣❛r✱ = −1, s✐ n ❡s − n n ✐♠♣❛r✳ Pr♦❜❧❡♠❛ ✶✳✸✳ ❙❡❛ Ω = {1, 2, 3, 4}✳ ❉❡s❝r✐❜✐r ❡①♣❧í❝✐t❛♠❡♥t❡ ❧♦s ❝♦♥❥✉♥t♦s P(Ω) ②2 A = : i + j  {(i, j) ∈ Ω ≤ 4}✳) A Pr♦❜❧❡♠❛ ✶✳✹✳ ❙✐ (A n n ❡s ✉♥❛ s✉❝❡s✐ó♥ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ② A = ∪ n n ✱ ❡♥t♦♥❝❡sn A = B = AC n n n k 1 2 n n ∪ ✱ ❞♦♥❞❡ B ∪ k=1 ✱ ② B ⊂ B ⊂ . , n2 } t❛❧ q✉❡ ≤ X(ω) < 2 k 2k2k−2 k−1 2k−2 2k−1 (ω) := (ω) , n n +1 n +1 n n n+1 n +1 n +1 ✱ ❡♥t♦♥❝❡s ≤ X(ω) < ② X ≤ X ∈ { }✳ ❙✐ 2 2 2 2 2 2 X(ω) (ω) := n (ω) (ω) = n + 1 n n+1 n+1 ≥ n ❡♥t♦♥❝❡s X ≤ X ♣✉❡s X s✐ X(ω) ≥ n + 1 ♦ nn n (2n+k)22(n+1)2 2n2n+1 X (ω) n +1 : k = 0, 1, . P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✷, ) i i 1 1 1 2 Pr♦❜❧❡♠❛ ✷✳✶✳ ❙❡❛♥ (Ω A ✱ i = 1, 2✱ ❡s♣❛❝✐♦s ♠❡❞✐❜❧❡s✱ A ∈ A ② X : A −→ Ω cc ′ : A 2 1 1 ② X 1 −→ Ω ❢✉♥❝✐♦♥❡s ♠❡❞✐❜❧❡s ♣❛r❛ ❧❛s σ✲á❧❣❡❜r❛s q✉❡ A ✐♥❞✉❝❡ ❡♥ A ② A 1 ✳ c′ ❊♥t♦♥❝❡s✱ ❧❛ ❛♣❧✐❝❛❝✐ó♥ S : Ω 1 2 q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♥ X ❡♥ A 1 ② ❝♦♥ X ❡♥ A ❡s −→ Ω 1✶ ♠❡❞✐❜❧❡✳ Pr♦❜❧❡♠❛ ✷✳✷✳ ❛✮ ❙❡❛♥ X, Y : (Ω, A) −→ (R, R) ❞♦s ✈✳❛✳r✳ Pr♦❜❛r q✉❡ {X = Y } ∈ A✱ {X 6= Y } ∈ A✱ {X < Y } ∈ A ② {X > Y } ∈ A✳ n ❜✮ ❊♥ ❝❛❞❛ ✉♥♦ ❞❡ ❧♦s ❛♣❛rt❛❞♦s s✐❣✉✐❡♥t❡s✱ ❞❡❝✐❞✐r s✐ ❡❧ ❝♦♥❥✉♥t♦ A ⊂ R ❡s ✉♥P n 2 2 n 2 : y = x : x < 1❜♦r❡❧✐❛♥♦✿ ❜✳✶✮ A = {(x, y) ∈ R }✳ ❜✳✷✮ A = {x ∈ R }✳ ❜✳✸✮ i=1 i 2 ∞ A = Z ]k× Z ⊂ R ✳ ❜✳✹✮ A = ∪ k=1 − 1/k, k + 1/k[⊂ R✳ ′′ Pr♦❜❧❡♠❛ ✷✳✸✳ ❙❡❛♥ Ω ✉♥ ❝♦♥❥✉♥t♦✱ (Ω, A) ✉♥ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ ② X : Ω −→ Ω ✉♥❛′ −1 : X (B)❛♣❧✐❝❛❝✐ó♥✳ Pr♦❜❛r q✉❡ ❧❛ ❢❛♠✐❧✐❛ {B ⊂ Ω ∈ A} ❡s ❧❛ ♠❛②♦r σ✲á❧❣❡❜r❛ ❡♥ ′ Ω q✉❡ ❤❛❝❡ ♠❡❞✐❜❧❡ ❛ X✳ n ) nPr♦❜❧❡♠❛ ✷✳✹✳ ❙❡❛ (X ✉♥❛ s✉❝❡s✐ó♥ ❞❡ ❢✉♥❝✐♦♥❡s ¯R✲✈❛❧♦r❛❞❛s ② ❇♦r❡❧✲♠❡❞✐❜❧❡s✳ X (ω) n n Pr♦❜❛r q✉❡ {ω ∈ Ω: l´ım ❡①✐st❡ ② ❡s ✜♥✐t♦} ❡s ✉♥ ❝♦♥❥✉♥t♦s ♠❡❞✐❜❧❡✳ n X Pr♦❜❧❡♠❛ ✷✳✺✳ ❈❛❧❝✉❧❛r sup n ✱ s✐❡♥❞♦ ( n −n = 2 x s✐ 0 ≤ x ≤ 2X : x (x) = n n ∈ [0, 1] −→ X −n = 1 < x s✐ 2 ≤ 1 ′ ) iPr♦❜❧❡♠❛ ✷✳✻✳ ❙❡❛♥ X : Ω −→ Ω ✉♥❛ ❛♣❧✐❝❛❝✐ó♥✱ (A i∈I ✉♥❛ ❢❛♠✐❧✐❛ ❞❡ s✉❜✲ ′′ )❝♦♥❥✉♥t♦s ❞❡ Ω ② (A i∈I ✉♥❛ ❢❛♠✐❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ✳ Pr♦❜❛r ❧❛s s✐❣✉✐❡♥t❡s i ♣r♦♣♦s✐❝✐♦♥❡s✿ −1 ′ −1 ′ ( A ) = X (A ) i i ✭❛✮ X ∪ i ∪ i ✳ −1 ′ −1 ′ ( i A ) = i X (A )✭❜✮ X ∩ ∩ ✳ i i A ) = X(A ) i i i i ✭❝✮ X(∪ ∪ ✳A ) X(A ) i i i i ✭❞✮ X(∩ ⊂ ∩ ✳ ▲❛ ✐❣✉❛❧❞❛❞ ♥♦ s✐❡♠♣r❡ s❡ ❞❛ ❡♥ ❡st❡ ❝❛s♦✳ EXc ′ ′ −1 ′c −1 ′ (A ) = X (A )✭❡✮ ❙✐ A ⊂ Ω ❡♥t♦♥❝❡s X ✳ U S 2 LEPr♦❜❧❡♠❛ ✷✳✼✳ ❈♦♥s✐❞❡r❡♠♦s ❧❛s ❛♣❧✐❝❛❝✐♦♥❡s X : (x, y) ∈ R −→ X(x, y) := x+y ∈ A 2 2 2 −1 −1 R + y ([0, 1]) (]0, 1])❡ Y : (x, y) ∈ R −→ X(x, y) := x ∈ R✳ ❉❡t❡r♠✐♥❛r X ❡ Y ✳ U N✶ c AA 1 ② A ❝♦♥st✐t✉②❡♥ ✉♥❛ ♣❛rt✐❝✐ó♥ ♠❡❞✐❜❧❡ ❞❡ Ω✱ ❡s ❞❡❝✐r✱ s♦♥ s✉❝❡s♦s ❞✐s❥✉♥t♦s q✉❡ r❡❝✉❜r❡♥1 M  ≥ − n A Ai i ❊♥t♦♥❝❡s✱ ❞❡ ❛❝✉❡r❞♦ ❝♦♥ ❧♦ ♣r♦❜❛❞♦ ♣❛r❛ ❢✉♥❝✐♦♥❡s s✐♠♣❧❡s✱ n Z Z Z X n λ( A ) = hdµ sdµ = sdµ.i ∪ i=1 ≥ n nA A A ∪ i ∪ i ii =1 i =1i=1 P♦r t❛♥t♦✱ n n Z X X n λ( A ) hdµ λ(A ) i i ∪ i=1 ≥ − ǫ = − ǫ. ≥ − l´ım inf −f n n Mn n ✺✶ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD❊❖❘❊▼❆ 1 ,❚ ✹✳✺✳ ✭❚❡♦r❡♠❛ ❞❡ ❧❛ ❝♦♥✈❡r❣❡♥❝✐❛ ❞♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡✮ ❙❡❛♥ g, f, f f , . P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✹  . , n}✱ A = P(Ω) ② P ❧❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❡♥ (Ω, A) ❞❡✜♥✐❞❛ ♣♦r n k n−k P ( p (1 , k = 0, 1, . 2 E(  (Ω, A, µ; R)✱ ❞❡♥♦t❛r❡♠♦s kfk p =Z Ω |f(ω)| p dµ(ω) 1/p . Pr♦❜❛r q✉❡ kf + gk p = kfk p p p n=1 p . E((X  i=1 2 ≤ P (X ≤ m)✮✱ ❡♥t♦♥❝❡s ♣❛r❛ ❝❛❞❛ a ∈ R✱E( |X − µ|) ≤ E(|X − a|)❊♥ ♣❛rt✐❝✉❧❛r✱ s✐ x 1 , . . 2 UN A−2 M P ( . |X − µ| ≥ kσ) ≤ k ✻✶ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD R R k k k k P ( α dP dP )❉❡♠♦str❛❝✐ó♥✳ ✭❛✮ α |X| ≥ α) = ≤ |X| ≤ E(|X| ✳ {|X|≥α} {|X|≥α} ✭❜✮ ❊s ❝♦♥s❡❝✉❡♥❝✐❛ ✐♥♠❡❞✐❛t❛ ❞❡ ✭❛✮✳ 2 (Ω, ❊❋■◆■❈■Ó◆❉ ✺✳✷✳ ✭❈♦✈❛r✐❛♥③❛✱ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛❝✐ó♥✮ ❙✐ X, Y ∈ L A, µ) s❡ ❞❡✜♥❡ ❈♦✈(X, Y ) = E[(X − EX)(Y − EY )]✳ ❊❧ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛❝✐ó♥ ❡♥tr❡ X (X,Y ) ❈♦✈❡ Y s❡ ❞❡✜♥❡ ❝♦♠♦ ρ(X, Y ) = ✳ (X) (Y ) √❱❛r √❱❛r ❖❜s❡r✈❛❝✐ó♥ ✺✳✸✳ ▲❛ ❞❡s✐❣✉❛❧❞❛❞ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ♣r✉❡❜❛ q✉❡ ❧❛ ❝♦✈❛r✐❛♥③❛ ❡stá ❜✐❡♥ ❞❡✜♥✐❞❛✳ ❊s s❡♥❝✐❧❧♦ ♣r♦❜❛r q✉❡ ❈♦✈(X, Y ) = E(XY ) − E(X)E(Y )✳ P♦r ♦tr❛♣❛rt❡✱ ❡s❛ ♠✐s♠❛ ❞❡s✐❣✉❛❧❞❛❞ ♣r✉❡❜❛ q✉❡ −1 ≤ ρ(X, Y ) ≤ 1✳ EX U S LE A U N A M ❈❆P❮❚❯▲❖ ✺✳ ❙❯▼❆ ❉❊ ▼❊❉■❉❆❙✳ ▼❊❉■❉❆ ■▼❆●❊◆✳ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ✻✷ ❉■❙❚❘■❇❯❈■❖◆❊❙ ❉❊ P❘❖❇❆❇■▲■❉❆❉ P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✺ Pr♦❜❧❡♠❛ ✺✳✶✳ ❍❛❝✐❡♥❞♦ T (x) = −x✱ x ∈ R✱ ♣r♦❜❛r q✉❡ ✉♥❛ ❢✉♥❝✐ó♥ ♠❡❞✐❜❧❡ g : R−→ R ❡s ✐♥t❡❣r❛❜❧❡ ✭r❡s♣❡❝t♦ ❛ ❧❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡✮ s✐✱ ② só❧♦ s✐✱ ❧❛ ❢✉♥❝✐ó♥ h(x) = g( −x) ❧♦ ❡s✳ ❆♥á❧♦❣❛♠❡♥t❡✱ s✐ c ∈ R✱ g ❡s ✐♥t❡❣r❛❜❧❡ s✐✱ ② só❧♦ s✐✱ g(x + c) ❧♦❡s✳ ❙❡ ✈❡r✐✜❝❛ ❛❞❡♠ás q✉❡ Z Z Z g(x)dx = g( g(x + c)dx. −x)dx = Pr♦❜❧❡♠❛ ✺✳✷✳ ❉❡t❡r♠✐♥❛r ❧❛ ❞✐str✐❜✉❝✐ó♥ ❞❡ ✉♥❛ ✈✳❛✳ r❡s♣❡❝t♦ ❛ ✉♥❛ ♠❡❞✐❞❛ ❞❡ ❉✐r❛❝✳ 6 p ε Pr♦❜❧❡♠❛ ✺✳✸✳ ❙❡❛♥ Ω = {1, 2, 3, 4, 5, 6}✱ A = P(Ω) ② P = P i i ✭❞♦♥❞❡i=1 P p p = 1 i i ≥ 0 ② ✮✳ ❈♦♥s✐❞❡r❡♠♦s ❧❛ ✈✳❛✳r T ❞❡✜♥✐❞❛ ♣♦r T (1) = T (2) = T (3) = −1✱ i T 2 T T (4) = T (5) = 0(f ) +x+1 ② T (6) = 1✳ ❉❡t❡r♠✐♥❛r P ✱ E(T ) ② E P ✱ ❞♦♥❞❡ f(x) = x ✳P 6 1 ε kPr♦❜❧❡♠❛ ✺✳✹✳ ❙❡❛ P = ✳ Pr♦❜❛r q✉❡ P ❡s ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❡♥ R✳ ❈❛❧❝✉❧❛r 6 k=1 √P (N), P ([ 2 + N) −2, 2]\] − ∞, 1]), P ( ② P (Z△[−3, 3]). ⋆Pr♦❜❧❡♠❛ ✺✳✺✳ ✭ ✮ ❙❡❛♥ X ✉♥❛ ✈❛r✐❛❜❧❡ ❛❧❡❛t♦r✐❛ r❡❛❧ ❡♥ ✉♥ ❡s♣❛❝✐♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞ (Ω,A, P ) ② ② g : R −→ R ✉♥❛ ❢✉♥❝✐ó♥ ❝r❡❝✐❡♥t❡ ♥♦ ♥❡❣❛t✐✈❛ t❛❧ q✉❡ E(g(X)) < ∞✳  , n} ❡s ❧❛ ❞✐str✐❜✉❝✐ó♥ ✳ ❉❡t❡r♠✐♥❛r s✉ ♠❡❞✐❛ ② s✉ i=1 n✶ ✈❛r✐❛♥③❛✳✭ ✮ Pr♦❜❧❡♠❛ ✺✳✶✵✳ ✭❈♦♠❜✐♥❛t♦r✐❛✮ ❘❡❝♦r❞❡♠♦s q✉❡✱ s✐ n ∈ N ② k ∈ {0, 1, . , n} q✉❡ t✐❡♥❡♥ k ❡❧❡♠❡♥t♦s✳ ❊s ♦❜✈✐♦ k n nk k n−k = q✉❡ ✳ ❙✐ k, n ∈ N✱ ❧♦s ❡❧❡♠❡♥t♦s ❞❡❧ ❝♦♥❥✉♥t♦ ♣r♦❞✉❝t♦ {1, . 1 I   , p , p ε 1 21 2 n x {x } ✉♥ s✉❜❝♦♥❥✉♥t♦ ♥✉♠❡r❛❜❧❡ ❞❡ R✱ p · · · ≥ 0 ② P = n ✳ P n≥1 n p n = 1 s❡rá ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ s✐ P ✳ ❙✐ X : R → R ❡s ✉♥❛ ✈✳❛✳r✳ ♥♦ ♥❡❣❛t✐✈❛✱ ❡♥t♦♥❝❡s X E (X) = p X(x ). , b m = b > r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙✐ π ❡s ♦tr❛ ♣❛rt✐❝✐ó♥ ❞❡ [a, b] ② ′ , a , . 1 C② t✐❡♥❡ ✐♥✈❡rs❛ ❞❡ ❝❧❛s❡ C ✮ ❞❡ ♦tr❛ ✈✳❛✳ n✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥ ❞❡♥s✐❞❛❞✳ n❈♦r♦❧❛r✐♦ ✻✳✶✳ ✭❚r❛♥s❢♦r♠❛❝✐ó♥ ❞❡ ✈❛r✐❛❜❧❡s✮ ❙❡❛♥ U ② V ❛❜✐❡rt♦s ❞❡ R ✱ T : U → V 1 −1 ✉♥❛ ❜✐②❡❝❝✐ó♥ t❛❧ q✉❡ T ② T s♦♥ ❞❡ ❝❧❛s❡ C ✱ ② X : (Ω, A, P ) → U ✉♥❛ ✈✳❛✳ n✲ n X ❞✐♠❡♥s✐♦♥❛❧ ❝♦♥ ❞❡♥s✐❞❛❞ f ✭r❡s♣❡❝t♦ ❛ ❧❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ λ ❡♥ R ✮✳ ❊♥t♦♥❝❡s❧❛ ✈✳❛✳ Y := T ◦ X t✐❡♥❡ ❞❡♥s✐❞❛❞ T ◦X dP −1− 1 f Y (y) = (y) = f (T (y)) (y) |J T |, y ∈ V.dλ n (V )❉❡♠♦str❛❝✐ó♥✳ ❙✐ B ∈ R ✱ ❡♥t♦♥❝❡s Z T ◦X −1 P (B) = P (T B) = f (x)dx = ( ◦ X ∈ B) = P (X ∈ T ∗). − 1 T B−1 (y)❍❛❝✐❡♥❞♦ ❡❧ ❝❛♠❜✐♦ x = T ✱ s❡ ❞❡❞✉❝❡ ❞❡❧ t❡♦r❡♠❛ ❞❡ ❝❛♠❜✐♦ ❞❡ ✈❛r✐❛❜❧❡s ✭❚❡♦✲ r❡♠❛ ✻✳✷ ✮ q✉❡ Z −1− 1 ( f (T (y)) (y) T ∗) = |J |dy, B ❧♦ q✉❡ ❛❝❛❜❛ ❧❛ ♣r✉❡❜❛✳ ❖❜s❡r✈❛❝✐ó♥ ✻✳✶✳ ❈♦♥ ❧❛s ♥♦t❛❝✐♦♥❡s ❡ ❤✐♣ót❡s✐s ❞❡❧ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ s❡ t✐❡♥❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ ❜♦r❡❧✐❛♥♦ B ❞❡ V ✱Z Z T 1 − λ (B) = dx = (y) T |J |dy, − 1 T B B T − 1 ❧♦ q✉❡ ♣r✉❡❜❛ q✉❡ ❧❛ ♠❡❞✐❞❛ ✐♠❛❣❡♥ λ t✐❡♥❡ ❞❡♥s✐❞❛❞ |J T | r❡s♣❡❝t♦ ❛ λ✳ EX U S LE A U N A M ∈ [0, 1)❡♥ ♦tr♦ ❝❛s♦ f (x) = s✐ x ≤ 0e Pr♦❜❧❡♠❛ ✻✳✸✳ ❙❡❛ X ✉♥❛ ✈❛r✐❛❜❧❡ ❛❧❡❛t♦r✐❛ ❝♦♥ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞ Z+2 . ② T = e 2 ❉❡t❡r♠✐♥❛r✱ ❧❛ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞ ❞❡ ❧❛s ✈❛r✐❛❜❧❡s ❛❧❡❛t♦r✐❛s S ② T ❞♦♥❞❡S = Z ∈ (−1, 0)−z + 1 s✐ z s✐ x > 0✶✳ ❉❡t❡r♠✐♥❛r ❧❛ ❢✉♥❝✐ó♥ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞ ❞❡ ❧❛ ✈❛r✐❛❜❧❡ ❛❧❡❛t♦r✐❛ Y = [X] ✭♣❛rt❡ f (z) = z + 1s✐ z Pr♦❜❧❡♠❛ ✻✳✷✳ ❙❡❛♥ Z ✉♥❛ ✈❛r✐❛❜❧❡ ❛❧❡❛t♦r✐❛ ❝♦♥ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞✿ dx✱ s✐ A ∈ A✳ x e A dx② µ(A) = R −x ❡♥t❡r❛ ❞❡ X✮✷✳ ❉❡t❡r♠✐♥❛r ❧❛ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞ ❞❡ Z = 3x R ✮ ✭■♥t❡❣r❛❧❡s ❞❡ ❘✐❡♠❛♥♥ ② ❞❡ ▲❡❜❡s❣✉❡✮ ❊♥ ❡st❡ ♣r♦❜❧❡♠❛ s❡♣r❡t❡♥❞❡♥ ❥✉st✐✜❝❛r ❧❛s ❛✜r♠❛❝✐♦♥❡s ❞❡❧ ❡❥❡♠♣❧♦ ✹ ❞❡ ❧❛ ♣á❣✐♥❛ ✻✾ ✳ ❈♦♥ ❧❛s ❞❡✜♥✐❝✐♦♥❡s② ♥♦t❛❝✐♦♥❡s ❛❧❧í ✉t✐❧✐③❛❞❛s✱ ♣r♦❜❛r✿ Pr♦❜❧❡♠❛ ✻✳✺✳ ✭⋆ ❞❡ ✉♥❛ ❞✐str✐❜✉❝✐ó♥ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞ ❡♥ R q✉❡ ❧❧❛♠❛r❡♠♦s ❞✐str✐❜✉❝✐ó♥ ❞❡ ❈❛✉❝❤② ❞❡♣❛rá♠❡tr♦ α✳ Pr♦❜❛r q✉❡ ❡s❛ ❞✐str✐❜✉❝✐ó♥ ♥♦ t✐❡♥❡ ♠❡❞✐❛ ✜♥✐t❛✳ = 1✳ ❊s ❧❛ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞ α c )❡s ✉♥❛ ❢✉♥❝✐ó♥ ❝♦♥t✐♥✉❛ ♥♦ ♥❡❣❛t✐✈❛ t❛❧ q✉❡ R X 1 + X 2 2 π(α (x) :=α α c Pr♦❜❧❡♠❛ ✻✳✹✳ ✭❉✐str✐❜✉❝✐ó♥ ❞❡ ❈❛✉❝❤②✮ Pr♦❜❛r q✉❡✱ s✐ α > 0✱ 2 A AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDADM A N U A LE S U EX 2 3 1 (ε 2 1 )② µ = 1 ✷✳ Ω = {1, 2, 3}✱ A = P(Ω)✱ λ = (ε 2 1 ◆✐❦♦❞②♠ ❞❡ λ r❡s♣❡❝t♦ ❛ µ ②✱ ❡♥ s✉ ❝❛s♦✱ ❝❛❧❝✉❧❛r❧❛✳✶✳ Ω = {1, 2, 3}✱ A = P(Ω)✱ λ = P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✻ Pr♦❜❧❡♠❛ ✻✳✶✳ ❊♥ ❧♦s ❛♣❛rt❛❞♦s s✐❣✉✐❡♥t❡s✱ ❞❡❝✐❞✐r s✐ ❡①✐st❡ ❧❛ ❞❡r✐✈❛❞❛ ❞❡ ❘❛❞♦♥✲ ❚❊❖❘❊▼❆ ❉❊ ❈❆▼❇■❖ ❉❊ ❱❆❘■❆❇▲❊❙ ✼✷❈❆P❮❚❯▲❖ ✻✳ ▼❊❉■❉❆❙ ❉❊❋■◆■❉❆❙ P❖❘ ❉❊◆❙■❉❆❉❊❙✳ )✳ 1 ✹✳ Ω = [0, 1]✱ A = R([0, 1])✱ λ(A) = R 2 dx s✐ A ∈ A ② µ ❧❛ ❞✐str✐❜✉❝✐ó♥ ✉♥✐❢♦r♠❡❡♥ [0, 1]✳ 2 3x A ✸✳ Ω = [0, 1]✱ A = R([0, 1])✱ λ(A) = R )✳ 3 1 2 (ε 3 1 )② µ = 2 1 (ε ✼✸ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD′ ′ ′ ) )✶✳ l(X, π) ≤ l(X, π ≤ u(X, π ≤ u(X, π) s✐ π ❡s ✉♥❛ ♣❛rt✐❝✐ó♥ ❞❡ [a, b] ♠ás ✜♥❛ q✉❡ π✳ ■♥❞✐❝❛❝✐ó♥✿ Pr♦❝❡❞❡r ♣♦r ✐♥❞✉❝❝✐ó♥✱ ❛ñ❛❞✐❡♥❞♦ ❧♦s ♣✉♥t♦s ❡①tr❛ ✉♥♦ ❛ ✉♥♦ ❛ ❧❛ ♣❛rt✐❝✐ó♥ π✳ ✷✳ ❯♥❛ ❢✉♥❝✐ó♥ ❛❝♦t❛❞❛ X : [a, b] → R ❡s ❘✐❡♠❛♥♥✲✐♥t❡❣r❛❜❧❡ s✐✱ ② só❧♦ s✐✱ ♣❛r❛❝❛❞❛ ǫ > 0✱ ❡①✐st❡ ✉♥❛ ♣❛rt✐❝✐ó♥ π t❛❧ q✉❡ u(X, π) − l(X, π) < ǫ✳ ✸✳ ❚♦❞❛ ❢✉♥❝✐ó♥ ❝♦♥t✐♥✉❛ ❡♥ [a, b] ❡s ❘✐❡♠❛♥♥✲✐♥t❡❣r❛❜❧❡✳ ■♥❞✐❝❛❝✐ó♥✿ ❯tí❧✐❝❡s❡ q✉❡ t♦❞❛ ❢✉♥❝✐ó♥ ❝♦♥t✐♥✉❛ ❡♥ ✉♥ ❝♦♠♣❛❝t♦ ❡s ✉♥✐❢♦r✲ ♠❡♠❡♥t❡ ❝♦♥t✐♥✉❛✳ ✹✳ ❯♥❛ ❢✉♥❝✐ó♥ ❛❝♦t❛❞❛ X ❡♥ [a, b] ❡s ❘✐❡♠❛♥♥✲✐♥t❡❣r❛❜❧❡ s✐✱ ② só❧♦ s✐✱ ❡❧ ❝♦♥❥✉♥t♦❞❡ ♣✉♥t♦s ❞❡ ❞✐s❝♦♥t✐♥✉✐❞❛❞ ❞❡ X t✐❡♥❡ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✳ ❊♥ ❡s❡ ❝❛s♦✱❧❛s ✐♥t❡❣r❛❧❡s ❞❡ ❘✐❡♠❛♥♥ ② ❞❡ ▲❡❜❡s❣✉❡ ❞❡ X ❡♥ [a, b] ❝♦✐♥❝✐❞❡♥✳ n ✭♠ás ✜♥❛ q✉❡ π ✮■♥❞✐❝❛❝✐ó♥✿ ✭=⇒✮ P❛r❛ ❝❛❞❛ n✱ ❡❧❡❣✐r ✉♥❛ ♣❛rt✐❝✐ó♥ π n−1 ) ) < 1/nt❛❧ q✉❡ u(X, π n n ✳ ❉❡✜♥✐r ❢✉♥❝✐♦♥❡s s✐♠♣❧❡s s n ② t n ❡♥ − l(X, π[a, b] := ´ınf X q✉❡ ❝♦✐♥❝✐❞❡♥ ❝♦♥ X ❡♥ ❡❧ ♣✉♥t♦ a ② ✈❛❧❡♥ m i ]a ,a ] ②i− 1 i M := sup X =< a , a , . . . , a >i ✱ r❡s♣✳✱ s✐ π n 1 k ✳ ❊♥t♦♥❝❡s s n ]a ,a ] n ≤ X ≤i− 1 ib bt s = l(X, π ) t = u(X, π ) )n ❡ R n n ❡ R n n ✳ (s n ❡s ✉♥❛ s✉❝❡s✐ó♥ ❝r❡❝✐❡♥t❡ ❛ a a ) ✉♥❛ ❢✉♥❝✐ó♥ s ② (t n ❡s ✉♥❛ s✉❝❡s✐ó♥ ❞❡❝r❡❝✐❡♥t❡ ❛ ✉♥❛ ❢✉♥❝✐ó♥ t✳ ❯s❛r ❡❧b (t t❡♦r❡♠❛ ❞❡ ❧❛ ❝♦♥✈❡r❣❡♥❝✐❛ ❞♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♣❛r❛ ♣r♦❜❛r q✉❡ R −as)dλ = 0 ❀ s✐❡♥❞♦ t ≥ s✱ ❞❡❜❡ s❡r t = s ❝✳s✳ ❡♥ [a, b]✳ ❙✐ s(x) = t(x) ②x ♥♦ ❡stá ❡♥ ♥✐♥❣✉♥❛ ❞❡ ❧❛s ♣❛rt✐❝✐♦♥❡s π n ✱ X ❡s ❝♦♥t✐♥✉❛ ❡♥ x✿ ❧✉❡❣♦ ❡❧❝♦♥❥✉♥t♦ ❞❡ ♣✉♥t♦s ❞❡ ❞✐s❝♦♥t✐♥✉✐❞❛❞ ❞❡ X t✐❡♥❡ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✳ ❆❞❡♠ás s ≤ X ≤ t ✐♠♣❧✐❝❛ q✉❡ X = s ❝✳s✳ ②✱ ♣♦r t❛♥t♦✱ X ❡s ▲❡❜❡s❣✉❡✲♠❡❞✐❜❧❡ ② ▲❡❜❡s❣✉❡✲✐♥t❡❣r❛❜❧❡ ② s✉ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡ ❝♦✐♥❝✐❞❡ ❝♦♥ ❧❛ ❞❡ s✱ ❡s ❞❡❝✐r✱ ❝♦♥ s✉ ✐♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥✳ ✭⇐=✮ ❙❡❛ X ❝♦♥t✐♥✉❛ ❝✳s✳n ❙❡❛ π n ✉♥❛ ♣❛rt✐❝✐ó♥ q✉❡ ❞✐✈✐❞❡ [a, b] ❡♥ 2 ✐♥t❡r✈❛❧♦s ❞❡ ✐❣✉❛❧ ❧♦♥❣✐t✉❞ ② ❝♦♥str✉②❛♠♦s ❛ ♣❛rt✐r ❞❡ ❡❧❧❛ ❢✉♥❝✐♦♥❡s s n ② t n ❝♦♠♦ ❛♥t❡s✳ ❊♥t♦♥❝❡sX(x) = l´ım s (x) = l´ım t (x) ❡♥ ❝❛❞❛ ♣✉♥t♦ x ❞❡ ❝♦♥t✐♥✉✐❞❛❞ ❞❡ x ②✱n n n n (u(X, π ) ♣♦r t❛♥t♦✱ ❝✳s✳ P♦r ❡❧ t❡♦r❡♠❛ ❞❡ ❧❛ ❝♦♥✈❡r❣❡♥❝✐❛ ❞♦♠✐♥❛❞❛✱ l´ım n n− l(X, π )) = 0n ✱ q✉❡ ♣r✉❡❜❛ q✉❡ X ❡s ❘✐❡♠❛♥♥✲✐♥t❡❣r❛❜❧❡✳ ✺✳ ✭❘❡❣❧❛ ❞❡ ❇❛rr♦✇✮ ❙❡❛♥ X ✉♥❛ ❢✉♥❝✐ó♥ ❛❝♦t❛❞❛ ② ❘✐❡♠❛♥♥✲✐♥t❡❣r❛❜❧❡ ❡♥ [a, b] ′ ② f ✉♥❛ ❢✉♥❝✐ó♥ ❞❡r✐✈❛❜❧❡ ❡♥ [a, b] t❛❧ q✉❡ X = f ✭❡s ❞❡❝✐r✱ f ❡s ✉♥❛ ♣r✐♠✐t✐✈❛ EXb X(x)dx = f (b)❞❡ X✮❀ ❡♥t♦♥❝❡s✱ R − f(a)✳ aU S , a , . . . , a > = LE ■♥❞✐❝❛❝✐ó♥✿ ❙❡❛♥ π =< a1 n ✉♥❛ ♣❛rt✐❝✐ó♥ ❞❡ [a, b]✱ m i A´ınf X = sup X ② M i ✳ P♦r ❡❧ t❡♦r❡♠❛ ❞❡ ❧♦s ✐♥❝r❡♠❡♥t♦s ✜♥✐✲[a 1 ,a ] [a ,a ]i− i i− 1 i U N, a [ t♦s✱ ♣❛r❛ ❝❛❞❛ i✱ ❡①✐st❡ t i i t❛❧ q✉❡ ∈]a i−1A M ′ f (a ) ) = f (t )(a ) = X(t )(a ).i i i i i − f(a i−1 − a i−1 − a i−1 ❈❆P❮❚❯▲❖ ✻✳ ▼❊❉■❉❆❙ ❉❊❋■◆■❉❆❙ P❖❘ ❉❊◆❙■❉❆❉❊❙✳ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ✼✹ ❚❊❖❘❊▼❆ ❉❊ ❈❆▼❇■❖ ❉❊ ❱❆❘■❆❇▲❊❙ ❊♥t♦♥❝❡sn n X X l(X, π) = m i (a i ) X(t i )(a i )− a i−1 ≤ − a i−1i=1 i=1 n X M (a ) = u(X, π).i i ≤ − a i−1i=1 P❡r♦n n X X X(t i )(a i ) = (f (a i ) )) = f (b)− a i−1 − f(a i−1 − f(a).i=1 i=1 ▲✉❡❣♦✱ ♣❛r❛ ❝❛❞❛ ♣❛rt✐❝✐ó♥ π✱ l(X, π) ≤ f(b)−f(a) ≤ u(X, π)✳ P✉❡st♦ q✉❡✱b X s✐ X ❡s ❘✐❡♠❛♥♥✲✐♥t❡❣r❛❜❧❡✱ t❛♠❜✐é♥ l(X, π) ≤ R ≤ u(X, π)✱ s❡ ❞❡❞✉❝❡aq✉❡✱ ❞❛❞♦ ǫ > 0✱ s✐ π ❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ✜♥❛✱ Z b X − [f(b) − f(a)] ≤ u(X, π) − l(X, π) < ǫ.a ✻✳ ❙✐ X ❡s ❛❝♦t❛❞❛ ❡♥ [a, b] ② ❝♦♥t✐♥✉❛ ❡♥ ✉♥ ♣✉♥t♦ c ∈]a, b[✱ ❧❛ ❛♣❧✐❝❛❝✐ó♥ f(x) :=R x ′ X (c) = X(c)❡s ❞❡r✐✈❛❜❧❡ ❡♥ ❡❧ ♣✉♥t♦ c ② f ✳ aEX U S LE A U N A M ▼❡❞✐❞❛ Pr♦❞✉❝t♦✳ ▼❡❞✐❞❛s ❞❡ ❚r❛♥s✐❝✐ó♥ ❙❡ ✐♥tr♦❞✉❝❡♥ ❛ ❝♦♥t✐♥✉❛❝✐ó♥ ❧❛s ♥♦❝✐♦♥❡s ❞❡ ♠❡❞✐❞❛ ❞❡ tr❛♥s✐❝✐ó♥ ✭② ♣r♦❜❛❜✐❧✐❞❛❞❞❡ tr❛♥s✐❝✐ó♥✮ ② ❞❡ ♠❡❞✐❞❛ ♣r♦❞✉❝t♦✳ ❆♠❜❛s ♥♦❝✐♦♥❡s ❥✉❡❣❛♥ ✉♥ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧❡♥ ❡st❛ ♦❜r❛ ✭♣✳❡❥✳✱ ❡♥ ❧♦s ❝♦♥❝❡♣t♦s ❞❡ ✐♥❞❡♣❡♥❞❡♥❝✐❛ ② ❞❡ ❞✐str✐❜✉❝✐ó♥ ❝♦♥❞✐❝✐♦♥❛❧✮✳▲❛ ♥♦❝✐ó♥ ❞❡ ♠❡❞✐❞❛ ❞❡ tr❛♥s✐❝✐ó♥ ♣❡r♠✐t❡ ♣r❡s❡♥t❛r ❧♦s t❡♦r❡♠❛s ❝❧ás✐❝♦s ❞❡ ❧❛♠❡❞✐❞❛ ♣r♦❞✉❝t♦ ② ❞❡ ❋✉❜✐♥✐ ❝♦♥ ♠❛②♦r ❣❡♥❡r❛❧✐❞❛❞ s✐♥ ✉♥ ❣r❛♥ ❡s❢✉❡r③♦ ❛❞✐❝✐♦♥❛❧✳ , ) , ) 1 1 2 2 ❊♥ ❧♦ q✉❡ s✐❣✉❡ (Ω A ② (Ω A s❡rá♥ ❡s♣❛❝✐♦s ♠❡❞✐❜❧❡s✳ ▲❛ s✐❣✉✐❡♥t❡ ❞❡✜♥✐❝✐ó♥❝♦♥str✉②❡ ❛ ♣❛rt✐r ❞❡ ❡❧❧♦s ✉♥ ♥✉❡✈♦ ❡s♣❛❝✐♦ ♠❡❞✐❜❧❡✳ Ω ❞❡♥♦t❛rá ❡❧ ❝♦♥❥✉♥t♦ ♣r♦❞✉❝t♦Ω 1 21 2 i × Ω q✉❡ s✉♣♦♥❞r❡♠♦s ♣r♦✈✐st♦ ❞❡ ❧❛ σ✲á❧❣❡❜r❛ ♣r♦❞✉❝t♦ A = A × A ✳ ❙✐ µ ❡s i ✉♥❛ ♠❡❞✐❞❛ ❡♥ A ✱ i = 1, 2✱ ❡st❛♠♦s ❛❤♦r❛ ✐♥t❡r❡s❛❞♦s ❡♥ ❝♦♥str✉✐r ✉♥❛ ♠❡❞✐❞❛ µ ❡♥) = µ (A )µ (A ) 1 2 1 1 2 2 ✭és❛ s❡rí❛ ✉♥❛ ♠❡❞✐❞❛ r❛③♦♥❛❜❧❡ ❡♥ ❡❧ ♣r♦❞✉❝t♦ A t❛❧ q✉❡ µ(A ×A❡♥ ❡❧ s❡♥t✐❞♦ ❞❡ q✉❡✱ ♣❛r❛ ❡❧❧❛✱ ✏❡❧ ár❡❛ ❞❡ ✉♥ r❡❝tá♥❣✉❧♦ ❡s ❧❛ ❧♦♥❣✐t✉❞ ❞❡ s✉ ❜❛s❡ ♣♦r❧❛ ❞❡ s✉ ❛❧t✉r❛✑✮✳ ❈♦♠♦ ✈❡r❡♠♦s✱ ♣♦❞❡♠♦s ❝♦♥s❡❣✉✐r ♠✉❝❤♦ ♠ás q✉❡ ❡st♦✳ ❊❋■◆■❈■Ó◆❉ ✼✳✶✳ ✭▼❡❞✐❞❛ ❞❡ tr❛♥s✐❝✐ó♥✮ ❯♥❛ ♠❡❞✐❞❛ ❞❡ tr❛♥s✐❝✐ó♥ ❡s ✉♥❛ ❛♣❧✐✲ 1 2 ❝❛❝✐ó♥ S : Ω × A −→ [0, +∞] t❛❧ q✉❡ EX U (ω ) = S(ω , A ) S 2 2 1 1 2 1 ✭✐✮ P❛r❛ ❝❛❞❛ A ∈ A ✱ S ❡s ✉♥❛ ❢✉♥❝✐ó♥ ❇♦r❡❧✲♠❡❞✐❜❧❡ ❞❡ ω ✳ LE (A ) = S(ω , A ) A ✭✐✐✮ P❛r❛ ❝❛❞❛ ω 1 1 ✱ S ω 1 2 1 2 ❡s ✉♥❛ ♠❡❞✐❞❛ ❡♥ A 2 ✳ ∈ Ω U N ❉✐r❡♠♦s q✉❡ ❧❛ ♠❡❞✐❞❛ ❞❡ tr❛♥s✐❝✐ó♥ S ❡s ✉♥✐❢♦r♠❡♠❡♥t❡ σ✲✜♥✐t❛ s✐ ❡①✐st❡♥ s✉❝❡✲ A M ) ) , B ) n 2 n 1 n n 1 1 s✐♦♥❡s (B ❡♥ A ② (k ❡♥ R t❛❧❡s q✉❡ S(ω ≤ k ✱ ♣❛r❛ ❝❛❞❛ ω ∈ Ω ② ❝❛❞❛ ✼✻ ❈❆P❮❚❯▲❖ ✼✳ ▼❊❉■❉❆ P❘❖❉❯❈❚❖✳ ▼❊❉■❉❆❙ ❉❊ ❚❘❆◆❙■❈■Ó◆ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ❡♥t❡r♦ n✳ S s❡ ❞✐❝❡ ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❞❡ tr❛♥s✐❝✐ó♥ s✐✱ ❛❞❡♠ás ❞❡ ✭✐✮✱ ✈❡r✐✜❝❛ q✉❡ ❧❛s, 1 2 ❢✉♥❝✐♦♥❡s ❞❡ ❝♦♥❥✉♥t♦ S(ω ·) s♦♥ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡♥ A ✳ ❊❖❘❊▼❆❚ ✼✳✶✳ ✭❚❡♦r❡♠❛ ❞❡ ❧❛ ♠❡❞✐❞❛ ♣r♦❞✉❝t♦ ❣❡♥❡r❛❧✐③❛❞♦✮ ❈♦♥s✐❞❡r❡♠♦s ❞♦si , i ) 1 1 ❡s♣❛❝✐♦s ♠❡❞✐❜❧❡s (Ω A ✱ i = 1, 2✱ ② s❡❛ µ ✉♥❛ ♠❡❞✐❞❛ σ✲✜♥✐t❛ ❡♥ A ✳ ❙❡❛ S ✉♥❛ 1 2 ♠❡❞✐❞❛ ❞❡ tr❛♥s✐❝✐ó♥ ✉♥✐❢♦r♠❡♠❡♥t❡ σ✲✜♥✐t❛ ❡♥ Ω × A ✳ ❊①✐st❡ ❡♥t♦♥❝❡s ✉♥❛ ú♥✐❝❛ 1 2 ♠❡❞✐❞❛ µ ❡♥ A = A × A t❛❧ q✉❡Z µ(A ) = S(ω , A )dµ (ω ), 1 2 1 2 1 1 × A A 1 1 1 2 2 ♣❛r❛ ❝❛❞❛ A ∈ A ② ❝❛❞❛ A ∈ A ✳ ❈♦♥❝r❡t❛♠❡♥t❡✱ s✐ A ∈ A✱ ❡♥t♦♥❝❡sZ µ(A) = S(ω , A(ω ))dµ (ω ), 1 1 1 1 Ω 1 ) = : (ω , ω ) 1 2 2 1 2 ❞♦♥❞❡ A(ω {ω ∈ Ω ∈ A}✳ ❆❞❡♠ás✱ µ ❡s σ✲✜♥✐t❛ ❡♥ A✳ P♦r ú❧t✐♠♦✱ s✐S 1 1 ❡s ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❞❡ tr❛♥s✐❝✐ó♥ ② µ ❡s ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❡♥ A ✱ ❡♥t♦♥❝❡s µ ❡s✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❡♥ A✳ , 1 ❉❡♠♦str❛❝✐ó♥✳ Pr✐♠❡r ❝❛s♦✿ ❙✉♣♦♥❣❛♠♦s ❡♥ ♣r✐♠❡r ❧✉❣❛r q✉❡ ❧❛s ♠❡❞✐❞❛s S(ω ·) s♦♥ ✜♥✐t❛s✳) ✭✐✮ Pr♦❜❡♠♦s q✉❡ s✐ A ∈ A✱ ❡♥t♦♥❝❡s A(ω 1 2 ✱ ∀ω 1 1 ✿ ❝♦♥s✐❞❡r❡♠♦s ❧❛ ∈ A ∈ Ω❢❛♠✐❧✐❛ ) 1 2 C := {A ∈ A: A(ω ∈ A }.c c A )(ω ) = A (ω ) (ω ) = A(ω ) n n 1 n n 1 1 1 C ❡s ✉♥❛ σ✲á❧❣❡❜r❛ ✭♣✉❡s (∪ ∪ ② A ✮ q✉❡ ❝♦♥t✐❡♥❡ ❛ )(ω ) =❧♦s r❡❝tá♥❣✉❧♦s ♠❡❞✐❜❧❡s ✭♣✉❡s✱ ♣❛r❛ ❝❛❞❛ A 1 1 ② ❝❛❞❛ A 2 2 ✱ (A 1 2 1 ∈ A ∈ A × AA 2 1 1 1 2 )(ω 1 ) = 1 / 1 s✐ ω ∈ A ② (A × A ∅ s✐ ω ∈ A ✮✳ P♦r t❛♥t♦✱ C = A✳, A(ω )) 1 1 1 ✭✐✐✮ Pr♦❜❡♠♦s ❛❤♦r❛ q✉❡✱ s✐ A ∈ A✱ S(ω ❡s ✉♥❛ ❢✉♥❝✐ó♥ A ✲♠❡❞✐❜❧❡✿ EX ❝♦♥s✐❞❡r❡♠♦s ❛❤♦r❛ ❧❛ ❢❛♠✐❧✐❛ US , A(ω )) 1 1 LE A U N 1 C := {A ∈ A: S(ω ❡s ✉♥❛ ❢✉♥❝✐ó♥ A ✲♠❡❞✐❜❧❡}. ❇❛st❛ ♣r♦❜❛r q✉❡ C = A✳ P❛r❛ ❡❧❧♦ ♣r♦❜❛r❡♠♦s q✉❡ C ❡s ✉♥ ❞✲s✐st❡♠❛ q✉❡ ❝♦♥t✐❡♥❡ A M ❛❧ π✲s✐st❡♠❛ ❢♦r♠❛❞♦ ♣♦r ❧♦s r❡❝tá♥❣✉❧♦s ♠❡❞✐❜❧❡s❀ ❡❧ t❡♦r❡♠❛ ❞❡ ❉②♥❦✐♥ ✭❚❡♦r❡♠❛ ✼✼ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD 1 1✶✳✶ ✮ ♣r♦❜❛rá q✉❡ C ❡s ✉♥❛ σ✲á❧❣❡❜r❛ ②✱ ♣♦r t❛♥t♦✱ q✉❡ C = A✳ P❡r♦✱ ❞❛❞♦s A ∈ A ② A 2 2 1 2 ∈ A ✱ s✐ A = A × A ✱ ❡♥t♦♥❝❡sS(ω 1 , A(ω 1 )) = S(ω 1 , A 2 )I A (ω 1 ), 1 1 1 2 q✉❡ ❡s ✉♥❛ ❢✉♥❝✐ó♥ A ✲♠❡❞✐❜❧❡✳ ❆❞❡♠ás C ❡s ✉♥ ❞✲s✐st❡♠❛ ♣✉❡s ❝♦♥t✐❡♥❡ ❛ Ω ×Ω ♣♦r s❡r ✉♥ r❡❝tá♥❣✉❧♦ ♠❡❞✐❜❧❡✱ ❡s ❡st❛❜❧❡ ❢r❡♥t❡ ❛ ❞✐❢❡r❡♥❝✐❛s ♣r♦♣✐❛s ✭♣✉❡s s✐ A, B ∈ C ②A 1 , (B 1 )) = S(ω 1 , B(ω 1 )) 1 , A(ω 1 )) 1 ⊂ B ❡♥t♦♥❝❡s S(ω \A)(ω −S(ω ✱ q✉❡ ❡s A ✲♠❡❞✐❜❧❡✱❧♦ q✉❡ ♣r✉❡❜❛ q✉❡ B \ A ∈ C✮ ② ❡s t❛♠❜✐é♥ ❡st❛❜❧❡ ❢r❡♥t❡ ❛ ❧❛ ✉♥✐ó♥ ♥✉♠❡r❛❜❧❡ ), ( A )(ω )) = n n1 n n 1 ❝r❡❝✐❡♥t❡ ✭♣✉❡s✱ s✐ (A ❡s ✉♥❛ s✉❝❡s✐ó♥ ❝r❡❝✐❡♥t❡ ❡♥ C ❡♥t♦♥❝❡s S(ω ∪ l´ım S(ω , A (ω ))A n 1 n 1 1 n n ✱ q✉❡ ❡s A ✲♠❡❞✐❜❧❡✱ ❧♦ q✉❡ ♣r✉❡❜❛ q✉❡ ∪ ∈ C✮✳✭✐✐✐✮ ❙❡ ❞❡✜♥❡ ✉♥❛ ❢✉♥❝✐ó♥ ❞❡ ❝♦♥❥✉♥t♦s ❡♥ ❧❛ σ✲á❧❣❡❜r❛ ♣r♦❞✉❝t♦ A ♣♦r Z µ(A) = S(ω , A(ω ))µ (dω ). 1 1 1 1 Ω 1 µ❡stá ❜✐❡♥ ❞❡✜♥✐❞❛ ♣♦r ✭✐✐✮✳ ❱❡❛♠♦s q✉❡ µ ❡s ✉♥❛ ♠❡❞✐❞❛ ❡♥ A q✉❡ ✈❡r✐✜❝❛ q✉❡ Zµ(A ) = S(ω , A )dµ (ω ), 1 2 1 2 1 1 × A A 1 1 1 2 2 ♣❛r❛ ❝❛❞❛ A ∈ A ② ❝❛❞❛ A ∈ A ✳ ❊s ❝❧❛r♦ q✉❡ µ ❡s ✉♥❛ ❢✉♥❝✐ó♥ ❞❡ ❝♦♥❥✉♥t♦s (n) ) n ♥♦ ♥❡❣❛t✐✈❛ ② ♥✉❧❛ ❡♥ ❡❧ ✈❛❝í♦✳ ❆❞❡♠ás✱ s✐ (A ❡s ✉♥❛ s✉❝❡s✐ó♥ ❞✐s❥✉♥t❛ ❡♥ A✱ s❡✈❡r✐✜❝❛ q✉❡ Z (n) (n) µ( A ) = S(ω , A (ω ))dµ (ω ) n 1 n 1 1 1 ∪ ∪ Ω1 Z  X (n) = S(ω , A (ω ))dµ (ω ) 1 1 1 1 Ω 1n X (n) = µ(A ).n ❊s♦ ♣r✉❡❜❛ q✉❡ µ ❡s ✉♥❛ ♠❡❞✐❞❛✳ P♦r ♦tr❛ ♣❛rt❡Z EX µ(A ) = S(ω , (A )(ω ))dµ (ω ) 1 2 1 1 2 1 1 1 × A × A UΩ Z 1 S LE = S(ω , A )I (ω )dµ (ω ) 1 2 A 1 1 1 1 AΩ Z 1 U N A = S(ω 1 , A 2 )dµ 1 (ω 1 ). , ω ) 1 n n 1 m ∈ B ⇐⇒ ∈ B m② ❞❡ q✉❡ Z ZQ (B ) = I (ω , . 1 C n  , ω n ),n +1 ≤ I n (1)(1) n = l´ım g n 1 n ❧♦ q✉❡ ♣r✉❡❜❛ q✉❡ g ❡s ✉♥❛ s✉❝❡s✐ó♥ ❞❡❝r❡❝✐❡♥t❡✳ ❙❡❛ h ✳ ❊❧ t❡♦r❡♠❛ ❞❡❧❛ ❝♦♥✈❡r❣❡♥❝✐❛ ♠♦♥ót♦♥❛ ❡①t❡♥❞✐❞♦ ♣r✉❡❜❛ q✉❡ Z l´ım P (C ) = h dP . n 1 1 nΩ 1′ ′ ′ P (C ) > 0 ) > 0 n n 1 P✉❡st♦ q✉❡ l´ım ✱ ❡①✐st❡ ω 1 t❛❧ q✉❡ h(ω 1 ❀ ❛❞❡♠ás✱ ω 1 ∈ B ♣✉❡s✱ ❡♥ ′ (ω , ω , . 2 EX  )②✱ ❝♦♠♦ ❛♥t❡s✱ (ω ∈ B ✳ P♦r ✐♥❞✉❝❝✐ó♥ ❝♦♥str✉✐♠♦s ✉♥ ♣✉♥t♦ (ω ∈ 1 2 1 2 U S Q ∞ ′ ′ ′ ′ Ω , . , n} × {0, 1} −→ [0, +∞[ q✉❡ ❡✈❛❧ú❛ ❡❧ ❝♦st❡ q✉❡♦❝❛s✐♦♥❛ ❧❛ t♦♠❛ ❞❡ ✉♥❛ ❞❡t❡r♠✐♥❛❞❛ ❞❡❝✐s✐ó♥ ❞❡❧ s✐❣✉✐❡♥t❡ ♠♦❞♦✿ ′ f (k, 1) = c, f (k, 0) = c k, k = 0, 1, . 2 N +y )2√ ∞ −xMe dx = π/2 ♣❛r❛ ♦❜t❡♥❡r ❞❡ ❛❤í q✉❡ R ✳ ✽✺ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD 1 2 √∞ x − 2 e dx = 2π Pr♦❜❧❡♠❛ ✼✳✺✳ (↑) ❈♦♥❝❧✉✐r q✉❡ R ✱ ❝♦♥ ❧♦ ❝✉❛❧ ❧❛ ❛♣❧✐❝❛❝✐ó♥−∞ 1 2 √ x −1 − 2 f : R 2π) e−→ R ❞❡✜♥✐❞❛ ♣♦r f(x) = ( ❡s ❧❛ ❞❡♥s✐❞❛❞ ❞❡ ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞ P ❡♥ (R, R)✳ ❊❧❧♦ ♥♦s ♣❡r♠✐t❡ ❞❛r ❧❛ s✐❣✉✐❡♥t❡ ❞❡✜♥✐❝✐ó♥✿ ❙❡❛ X ✉♥❛ ✈✳❛✳r✳ ❡♥ (Ω, A, P )❀❞✐r❡♠♦s q✉❡ X ❡s ✉♥❛ ✈✳❛✳r✳ ❝♦♥ ❞✐str✐❜✉❝✐ó♥ N(0, 1) ✭♥♦r♠❛❧ ❞❡ ♠❡❞✐❛ ❝❡r♦ ② ✈❛r✐❛♥③❛ X ✶✮ s✐ s✉ ❞✐str✐❜✉❝✐ó♥ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞ P ❡♥ (R, R) ❛❞♠✐t❡ ❝♦♠♦ ❞❡♥s✐❞❛❞ ✭r❡s♣❡❝t♦❛ ❧❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡✮ ❧❛ ❛♣❧✐❝❛❝✐ó♥ f q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❞❡✜♥✐r✳ Pr♦❜❧❡♠❛ ✼✳✻✳ (↑) ❙✐ X ❡s ✉♥❛ ✈✳❛✳r✳ ❝♦♥ ❞✐str✐❜✉❝✐ó♥ N(0, 1) ♣r♦❜❛r q✉❡ E(X) = 0✳∞ α−1 −x x e dx Pr♦❜❧❡♠❛ ✼✳✼✳ (↑) ❙❡❛♥ α > 0 ② Γ(α) := R ✭❡s ❧❛ ❧❧❛♠❛❞❛ ❢✉♥❝✐ó♥ ❣❛♠♠❛ ❞❡ ❊✉❧❡r✮✳ Pr♦❜❛r q✉❡ Γ(α) ❡stá ❜✐❡♥ ❞❡✜♥✐❞❛✱ ❡s ❞❡❝✐r✱ q✉❡ ❧❛ ✐♥t❡❣r❛❧ q✉❡ ❧❛❞❡✜♥❡ ❡①✐st❡ ② ❡s ✜♥✐t❛✳ ■♥❞✐❝❛❝✐ó♥✿ ▲❛ ✐♥t❡❣r❛❧ ❡①✐st❡ ♣✉❡s ❡❧ ✐♥t❡❣r❛♥❞♦ ❡s ✉♥❛ ❢✉♥❝✐ó♥ ♠❡❞✐❜❧❡ ♥♦−x/2 ♥❡❣❛t✐✈❛❀ ❛❝♦t❛r ❡❧ ✐♥t❡❣r❛♥❞♦ ♣♦r e ❡♥ ✉♥ ❝✐❡rt♦ ❡♥t♦r♥♦ ❞❡ +∞ ② ♣♦r ✉♥❛❢✉♥❝✐ó♥ ✐♥t❡❣r❛❜❧❡ ❡♥ ✉♥ ❡♥t♦r♥♦ ❞❡ ❝❡r♦ ✭s✐ α ≥ 1 ♥♦ ❤❛② ♣r♦❜❧❡♠❛s ❡♥ ❡st❡ α−1 −x α−1e ú❧t✐♠♦ ❝❛s♦✱ ② s✐ 0 < α < 1 s❡ ♣♦❞rá ❛❝♦t❛r x ♣❛r❛ x ♣ró①✐♠♦ ❛ ≤ x ❝❡r♦✮✳Pr♦❜❧❡♠❛ ✼✳✽✳ (↑) Γ(1) = 1✳ Pr♦❜❧❡♠❛ ✼✳✾✳ (↑) ❙✐ α > 1 ❡♥t♦♥❝❡s Γ(α) = (α − 1)Γ(α − 1)✳ ■♥❞✐❝❛❝✐ó♥✿ ■♥t❡❣r❛r ♣♦r ♣❛rt❡s✳ √ 1 ) = π Pr♦❜❧❡♠❛ ✼✳✶✵✳ (↑) Γ( ✳ 2 1 ∞ −−x 2) = x e dx ■♥❞✐❝❛❝✐ó♥✿ Γ( ❀ ❛♣❧✐❝❛r ❡❧ t❡♦r❡♠❛ ❞❡ ❝❛♠❜✐♦ ❞❡ ✈❛r✐❛❜❧❡s 2 2✉t✐❧✐③❛♥❞♦ ❧❛ tr❛♥s❢♦r♠❛❝✐ó♥ z ∈]0, +∞[−→ z ∈]0, +∞[✳ Pr♦❜❧❡♠❛ ✼✳✶✶✳ (↑) ❙✐ X t✐❡♥❡ ❞✐str✐❜✉❝✐ó♥ N(0, 1) ♣r♦❜❛r q✉❡ ❱❛r(X) = 1✳■♥❞✐❝❛❝✐ó♥✿ ◆♦t❛r q✉❡ Z 1 1 2 2 2 x − − 2 2] = x (2π) e dx ❱❛r(X) = E[(X − E(X)) −∞Z ∞ 1 1 22 x − − 2 2= 2(2π) x e dx; EX 2❤❛❝❡r ❛❤♦r❛ ❡❧ ❝❛♠❜✐♦ ❞❡ ✈❛r✐❛❜❧❡s y = x ② ✉t✐❧✐③❛r ❧♦s ❞♦s ♣r♦❜❧❡♠❛s ❛♥t❡r✐♦r❡s U S♣❛r❛ ❝❛❧❝✉❧❛r Γ(3/2)✳ LE(x−µ)2 A √ R 2σ2U 2π] e dx = Pr♦❜❧❡♠❛ ✼✳✶✷✳ (↑) ❙✐ µ ∈ R ② σ > 0 ♣r♦❜❛r q✉❡ [σ−∞ N ✳ ❉❡❢✳✿ ❙✐ X ❡s ✉♥❛ ✈✳❛✳r✳ ❝✉②❛ ❞✐str✐❜✉❝✐ó♥ q✉❡❞❛ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ❧❛ ❞❡♥s✐❞❛❞ (x−µ) √ M− 2 −12σ2 [σ 2π] e) ✱ ❞✐r❡♠♦s q✉❡ X ❡s ✉♥❛ ✈✳❛✳ N(µ, σ ✳ ✽✻ ❈❆P❮❚❯▲❖ ✼✳ ▼❊❉■❉❆ P❘❖❉❯❈❚❖✳ ▼❊❉■❉❆❙ ❉❊ ❚❘❆◆❙■❈■Ó◆ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD■♥❞✐❝❛❝✐ó♥✿ ❊❢❡❝t✉❛r ❡❧ ❝❛♠❜✐♦ ❞❡ ✈❛r✐❛❜❧❡s y = (x − µ)/σ✳ 2 ) Pr♦❜❧❡♠❛ ✼✳✶✸✳ (↑) Pr♦❜❛r q✉❡✱ s✐ X ❡s ✉♥❛ ✈✳❛✳ ❝♦♥ ❞✐str✐❜✉❝✐ó♥ N(µ, σ ✱ ❡♥t♦♥❝❡s2 E(X) = µ  i ) ✭❜✮ ▲♦s s✉❝❡s♦s ❞❡ ✉♥❛ ❢❛♠✐❧✐❛ (A i∈I ❞❡ A s❡ ❞✐❝❡♥ ✐♥❞❡♣❡♥❞✐❡♥t❡s s✐ ♣❛r❛ ❝❛❞❛Q n n , . . A, P ) −→ (Ω A ✱ i = 1, 2, . . . ✱ ✈✳❛✳ ▲❛ ❞✐str✐❜✉✲  X ❝✐ó♥ P ❞❡ ❧❛ ✈✳❛✳ EX U Q Q SX : ω 1 (ω), X 2 (ω), . . A, P ) → (Ω A ✱ 1 ≤ i ≤ n✱ ✈✳❛✳✱ ❞♦♥❞❡ µ ❡s  i i i i i i i i i i n ∩ {X ∈ A }) = ∈ A Q A Ai i ii i i=1 Q n f (ω )dµ(ω , . , x n ) = x 1 n d P (x i ) · · · x · · · x ii Q Q Xi = R x dP (x ) = E(X ).i i i i i , . EX )(X )]  )(x, y) =Z Z 2 1 ∗ P 2 ❞❡ P 1 ② P ♠❡❞✐❛♥t❡P ) 1 ∗ P 2 := (P 1 × P 2 2 I B−x(y)dP =P 2 . Z = P X 1 ∗ P X ❉❡♠♦str❛❝✐ó♥✳ ❙✐ X = (X ✱❡♥t♦♥❝❡s 1 , X 2 ) ✱ s✐❡♥❞♦ X 1 ② X 2 ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ s❡ t✐❡♥❡ q✉❡ P X P 2 2 Z P (y)dP 1 (x)= Z P 2 (B − x)dP 1 (x) = 1 (B 1 +X − y)dP 2 (y). H(Q)② ♣r♦❜❛r q✉❡✱ s✐ P ❡s ♦tr❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❡♥ Ω✱ ❡♥t♦♥❝❡s H(P ) < H(Q)✳ ✾✾ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD−1 −1 = log n > 0 ✳ ❙✐ P (k) = p k■♥❞✐❝❛❝✐ó♥✿ H(Q) = −n · n · log n ✱ 1 ≤ k ≤ n✱ ❞❡❜❡♠♦s ♣r♦❜❛r q✉❡n X p log pk k − ≤ log nk=1 ② ♣❛r❛ ❡❧❧♦ ✉t✐❧✐③❛r❡♠♦s ❡❧ ♠ét♦❞♦ ❞❡ ❧♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✿m ✏❙❡❛♥ m, n ∈ N t❛❧❡s q✉❡ m > n✱ U ❛❜✐❡rt♦⊂ R ② φ : U −→ Rn 1❢✉♥❝✐♦♥❡s ❞❡ ❝❧❛s❡ C ② f : U −→ R ❡♥ U✳ ❙❡❛♥ M = {x ∈ U :f (x) = 0 ❞❡ φ ❛ M } ② c ∈ M✳ ❙✉♣♦♥❣❛♠♦s q✉❡ ❧❛ r❡str✐❝❝✐ó♥ φ|Mt✐❡♥❡ ✉♥ ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❡♥ c ② q✉❡ ❡❧ r❛♥❣♦ ❞❡ ❧❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ ∂fi (c) ❡♥ ❡❧ ♣✉♥t♦ c ❡s n✳ ❊♥t♦♥❝❡s ❡①✐st❡ ✉♥ ú♥✐❝♦∂xj 1≤i≤n,1≤k≤m nλ = (λ , . . . , λ ) 1 n t❛❧ q✉❡∈ Rn X ∂φ ∂φ(c) + λ (c) = 0, 1i · ≤ j ≤ m. ∂x ∂xj j i=1 ❙✐ φ ② f ❛❞♠✐t❡♥ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛❧❡s ❝♦♥t✐♥✉❛s ❡♥ U ❡♥t♦♥❝❡s✱ ♣❛r❛ q✉❡ φ|M t❡♥❣❛ ✉♥ ♠í♥✐♠♦ ✭r❡s♣✳✱ ♠á①✐♠♦✮ r❡❧❛t✐✈♦ ❡♥ ❡❧ ♣✉♥t♦ c ∈ M2 L(c)(h, h) > 0  ❡ s✉✜❝✐❡♥t❡ q✉❡ D ✭r❡s♣✳✱ < 0✮ ❝❛❞❛ ✈❡③ q✉❡ h ∈m R (c)(h) = 0i \ {0} s❛t✐s❢❛❝❡ q✉❡ Df ✱ 1 ≤ i ≤ n✱ ❞♦♥❞❡ L = φ + Pn λ i f i ✳✑i=1 Pn n − ✱ U = {x ∈ Ri=1 , . , ω ) = ω ) ) )] ) 1 n 1 5 6 2 3 4 7 · 1 − (1 − ω · ω · (1 − ω · [1 − (1 − ω · (1 − ω · ω n , . A, P ) → N ❡s ✉♥❛ ✈✳❛✳ ② p ✱ n ≥ 1✱ s❡ ❞❡✜♥❡ s✉ ❢✉♥❝✐ó♥ ❣❡♥❡r❛tr✐③ ❞❡P X n ) = p z n ♣r♦❜❛❜✐❧✐❞❛❞ ❝♦♠♦ ❧❛ ❛♣❧✐❝❛❝✐ó♥ g(z) := E(z ✳ ❊st❛ s❡r✐❡ ❡s ❝♦♥✈❡r❣❡♥t❡ n ∞ ❡♥ [−1, 1] ② ❞❡✜♥❡ ✉♥❛ ❢✉♥❝✐ó♥ ❞❡ ❝❧❛s❡ C ❡♥ ] − 1, 1[✳ ❊s ♦❜✈✐♦ q✉❡ g ❝❛r❛❝t❡r✐③❛ ❧❛❞✐str✐❜✉❝✐ó♥ ❞❡ X✱ ♣✉❡s ❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡s❛ s❡r✐❡ ❞❡ ♣♦t❡♥❝✐❛s s❡ ♣✉❡❞❡♥ ♦❜t❡♥❡r ❛ (n) = n!g (0) n ♣❛rt✐r ❞❡ ❧❛s ❞❡r✐✈❛❞❛s s✉❝❡s✐✈❛s ❞❡ g ❡♥ ✵✿ ❝♦♥❝r❡t❛♠❡♥t❡✱ p ✳ ❙❡ ♣r✉❡❜❛✐♥❝❧✉s♦ q✉❡ X t✐❡♥❡ ♠♦♠❡♥t♦ ✜♥✐t♦ ❞❡ ♦r❞❡♥ p ≥ 1 s✐✱ ② só❧♦ s✐✱ ❡s ✜♥✐t♦ ❡❧ ❧í♠✐t❡♣♦r ❧❛ ✐③q✉✐❡r❞❛ ❡♥ ✶ ❞❡ ❧❛ ❞❡r✐✈❛❞❛ p✲és✐♠❛ ❞❡ g✳ ❊♥ ❡❧ ❝❛s♦ p = 1✱ s✐ X t✐❡♥❡ ♠❡❞✐❛ ′ g (z)✜♥✐t❛✱ E(X) = l´ım ✳ z→1− 2 ) ❖❜s❡r✈❛❝✐ó♥ ✾✳✾✳ ❉✐r❡♠♦s q✉❡ ✉♥❛ ✈✳❛✳r✳ X t✐❡♥❡ ❞✐str✐❜✉❝✐ó♥ N(µ, σ ✭✈❡r Pr♦✲ ❜❧❡♠❛ ✼✳✺ ② s✐❣✉✐❡♥t❡s✮ s✐ s✉ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞ ❡s 1 1 2 exp (x√ − − µ) 2 2σσ 2π ❙✉ ❢✉♥❝✐ó♥ ❝❛r❛❝t❡ríst✐❝❛ ❡s 1 2 2 ϕ (t) = exp itµ σ t , t X − ∈ R. 2 ❖❜s❡r✈❛❝✐ó♥ ✾✳✶✵✳ ❯♥❛ ✈✳❛✳r✳ X s❡ ❞✐❝❡ q✉❡ t✐❡♥❡ ❞✐str✐❜✉❝✐ó♥ ❣❛♠♠❛ ❞❡ ♣❛rá♠❡tr♦s α✱ β ✭α > 0✱ β > 0✮✱ ② s❡ ❡s❝r✐❜✐rá X ∼ G(α, β)✱ s✐ s✉ ❞✐str✐❜✉❝✐ó♥ ❛❞♠✐t❡ ❝♦♠♦ ❞❡♥s✐❞❛❞ ❧❛ ❛♣❧✐❝❛❝✐ó♥ α −1 α−1 −y/β [Γ(α)β ] y e I (y). ]0,+∞[ ❙✉ ❢✉♥❝✐ó♥ ❝❛r❛❝t❡ríst✐❝❛ ❡s 1ϕ (t) = , t α X ∈ R. (1− βit) EX ❈❛s♦s ❡s♣❡❝✐❛❧❡s ❞❡ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❣❛♠♠❛ s♦♥ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❝❤✐✲❝✉❛❞r❛❞♦ ❝♦♥ n n (n) := G( , 2)❣r❛❞♦s ❞❡ ❧✐❜❡rt❛❞ χ ✱ ② ❧❛ ❞✐str✐❜✉❝✐ó♥ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ♣❛rá♠❡tr♦ 2 U 2 S λ > 0✱ E(λ) := G(1, λ)✳ LE A U N A M ✶✶✶ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✾  {k}) = e k!❉❡t❡r♠✐♥❛r ❧❛ ❢✉♥❝✐ó♥ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❞❡ P♦✐ss♦♥ ② ♣r♦❜❛r q✉❡ s✐ PX , . , X ) X λ ) 1 n i i i i s♦♥ ✈✳❛✳ ✐♥❞❡♣❡♥❞✐❡♥t❡s ② X ∼ P (λ ❡♥t♦♥❝❡s P ∼ P ( ✳ i i , . 1 C =   . , ω n ) = P (ω 1 ) n ) = , · · · P (ω n N n n , . 1 F (x, ω) := I (X i (ω))  ]−∞,x] n i=1❖❜s❡r✈❛❝✐ó♥ ✶✵✳✹✳ n · F (x, ω) ❡s ❡❧ ♥ú♠❡r♦ ❞❡ í♥❞✐❝❡s i ∈ {1, . , n} t❛❧❡s q✉❡ X (ω) i ≤ x✳ ❖❜s❡r✈❛❝✐ó♥ ✶✵✳✺✳ ❉❛❞♦ ω ∈ Ω✱ F (·, ω) ❡s ❧❛ ❢✉♥❝✐ó♥ ❞❡ ❞✐str✐❜✉❝✐ó♥ ❞❡ ❧❛ ✈✳❛✳ ❞✐s❝r❡t❛Z : i (i) := X (ω) ωω i ∈ ({1, . 1 Q(Z (ω)  ω ❝❛r❞ {j ∈ {1, . . 2 E(S ) = E  X X i − n ¯ n− 1 i " # X 1 2 2 = E(X ) X ) i − nE( ¯ n− 1 i 1 1 2 2 = n( ) + E(X ) ) ) + E(X ) 1 1 1 1 ❱❛r(X − n ❱❛r(X n n− 1 n 1 = ) ) = ). . 2 N (µ, σ )A ✳ U 2 (n)✭❛✮ ✭❉✐str✐❜✉❝✐ó♥ χ ✮ ▲❛ ❞✐str✐❜✉❝✐ó♥ χ ✕❧❧❛♠❛❞❛ ❝❤✐✲❝✉❛❞r❛❞♦ ❝♦♥ n ❣r❛❞♦s 2 N A M ❞❡ ❧✐❜❡rt❛❞✕ ❡s ❧❛ ❞❡✜♥✐❝✐ó♥ ❣❛♠♠❛ G(n/2, 2)✳ ✶✷✸ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ✭❜✮ ✭❉✐str✐❜✉❝✐ó♥ t(n) ❞❡ ❙t✉❞❡♥t✮ ❙❡ ❞❡✜♥❡ ❧❛ ❞✐str✐❜✉❝✐ó♥ t(n) ❞❡ ❙t✉❞❡♥t ❝♦♥ n❣r❛❞♦s ❞❡ ❧✐❜❡rt❛❞ ❝♦♠♦ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❞❡❧ ❝♦❝✐❡♥t❡ X√ n , √Y 2 (n)❞♦♥❞❡ X ❡ Y s♦♥ ✈✳❛✳r✳ ✐♥❞❡♣❡♥❞✐❡♥t❡s t❛❧❡s q✉❡ X ∼ N(0, 1) ❡ Y ∼ χ ✳ ✭❝✮ ✭❉✐str✐❜✉❝✐ó♥ F ❞❡ ❋✐s❤❡r✮ ❙❡ ❞❡✜♥❡ ❧❛ ❞✐str✐❜✉❝✐ó♥ F (m, n) ❞❡ ❋✐s❤❡r ❝♦♥(m, n) ❣r❛❞♦s ❞❡ ❧✐❜❡rt❛❞ ❝♦♠♦ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❞❡❧ ❝♦❝✐❡♥t❡X/m ,Y /n 2 2 (m) (n)❞♦♥❞❡ X ❡ Y s♦♥ ✈✳❛✳r✳ ✐♥❞❡♣❡♥❞✐❡♥t❡s t❛❧❡s q✉❡ X ∼ χ ❡ Y ∼ χ ✳ 2 2 2 )σ ) ▲❡♠❛ ✶✵✳✶✳ ✭❛✮ ❙✐ X ∼ N(µ, σ ② a, b ∈ R ❡♥t♦♥❝❡s aX + b ∼ N(aµ + b, a ✳ 2 , . . . , X , σ )✭❜✮ ❙✐ X 1 n s♦♥ ✐♥❞❡♣❡♥❞✐❡♥t❡s ② X i ∼ N(µ i ✱ 1 ≤ i ≤ n✱ ❡♥t♦♥❝❡s i P P P n n n2 X i µ i , σ )  , X n Pr♦♣♦s✐❝✐ó♥ ✶✵✳✷✳ ❙❡❛ X ✉♥❛ ♠✉❡str❛ ❞❡ t❛♠❛ñ♦ n ❞❡ ✉♥❛ ❞✐str✐❜✉❝✐ó♥ 2 N (µ, σ ) ✳ 2 X /n) ✭❛✮ ¯ t✐❡♥❡ ❞✐str✐❜✉❝✐ó♥ N(µ, σ ✳ X X, . , t ) 1 n ( ¯ X,X X,...,X X)1 − ¯ n − ¯ = ϕ (t) (t , . 2 A (1)  . , X , Y , . 2 S /S  , 9} ② P Pn n −i −i 10 x = 10 y = yi i ❡♥t♦♥❝❡s x i i ✱ 1 ≤ i ≤ n✳i=1 i=1 Pr♦❜❧❡♠❛ ✶✵✳✷✶✳ ✭↑✮ ➽❈✉á❧ ❡s ❧❛ ❞✐str✐❜✉❝✐ó♥ ❞❡ ❧❛ ✈✳❛✳r✳ P ∞ ′−i X X . , 9}✱ ∀i✳ ❯t✐❧✐③❛♥❞♦ q✉❡i=1 P∞ −i ∞ 1 1 i=1 n=1 n−1x , X (ω) < xn n n−1 } ✭♣r✉é❜❡s❡ ❡s❛ ✐❣✉❛❧❞❛❞✮ ♣r♦❜❛r q✉❡ ❧❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❞❡ ❡s❡ 10 X (ω) < x (ω) = x , . 1 N −N  , p ) 1 r 1 r ❀ ❧❛ ❞❡♥♦t❛r❡♠♦s ♣♦r M(n; p ✳ r 1 , . 1 2 1 2 1 2 ≤ λ(A A A A 1 2 ) n n ✲ ❙❡❛ (f ✉♥❛ s✉❝❡s✐ó♥ ❡♥ F t❛❧ q✉❡Z Z l´ım f dµ = sup f dµ : f . P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❆P❮❚❯▲❖ ✶✶xPr♦❜❧❡♠❛ ✶✶✳✶✳ ❛✮ Pr♦❜❛r q✉❡ ❧❛ ♠❡❞✐❞❛ ❞❡ ❉✐r❛❝ ε ❡♥ ✉♥ ♣✉♥t♦ x ∈ R ♥♦ t✐❡♥❡ ❞❡♥s✐❞❛❞ r❡s♣❡❝t♦ ❛ ❧❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♥ R✳ ❜✮ ❈❛❧❝✉❧❛r ❧❛ ❞❡♥s✐❞❛❞ ❞❡ ✉♥❛❞✐str✐❜✉❝✐ó♥ N(µ, 1) r❡s♣❡❝t♦ ❛ ✉♥❛ ❞✐str✐❜✉❝✐ó♥ N(0, 1)✳ ❝✮ Pr♦❜❛r q✉❡ ❧❛ ❞✐str✐❜✉❝✐ó♥✉♥✐❢♦r♠❡ ❞✐s❝r❡t❛ ❡♥ {1, . . . , n} t✐❡♥❡ ❞❡♥s✐❞❛❞ r❡s♣❡❝t♦ ❛ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❜✐♥♦♠✐❛❧ b (1/2) n ② ❝❛❧❝✉❧❛r❧❛✳Pr♦❜❧❡♠❛ ✶✶✳✷✳ ❙❡❛♥ λ, µ ② ν ♠❡❞✐❞❛s σ✲✜♥✐t❛s ❡♥ (Ω, A) t❛❧❡s q✉❡ λ ≪ µ ≪ ν✳ Pr♦❜❛r q✉❡ dλ dλ dµ= , ν · − ❝✳s✳ dν dµ dν −1 ❊♥ ♣❛rt✐❝✉❧❛r✱ s✐ λ ≪ µ ② µ ≪ λ✱ ❡♥t♦♥❝❡s dλ/dµ = (dµ/dλ) ✱ µ✲❝✳s✳ Pr♦❜❧❡♠❛ ✶✶✳✸✳ ❙❡❛♥ (Ω, A, µ) ✉♥ ❡s♣❛❝✐♦ ❞❡ ♠❡❞✐❞❛ σ✲✜♥✐t♦ ② T : (Ω, A) → (Ω, A)✳T −1 (N )Pr♦❜❛r q✉❡ µ ≪ µ s✐✱ ② só❧♦ s✐✱ T ❡s ✉♥ s✉❝❡s♦ µ✲♥✉❧♦ ♣❛r❛ ❝❛❞❛ s✉❝❡s♦ µ✲♥✉❧♦N ✳ Pr♦❜❧❡♠❛ ✶✶✳✹✳ P♦♥❡r ✉♥ ❡❥❡♠♣❧♦ ❞❡ ❞♦s ✈✳❛✳r✳ X ❡ Y s♦❜r❡ ✉♥ ♠✐s♠♦ ❡s♣❛❝✐♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞ ❞❡ ❢♦r♠❛ q✉❡ ❝❛❞❛ ✉♥❛ ❞❡ ❡❧❧❛s t❡♥❣❛ ❞❡♥s✐❞❛❞ ✭r❡s♣❡❝t♦ ❛ ❧❛ ♠❡❞✐❞❛ ❞❡▲❡❜❡s❣✉❡ ❡♥ R✮✱ ♣❡r♦ q✉❡ (X, Y ) ♥♦ t❡♥❣❛ ❞❡♥s✐❞❛❞ r❡s♣❡❝t♦ ❛ ❧❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ 2 ❡♥ R ✳ Pr♦❜❧❡♠❛ ✶✶✳✺✳ ❙❡❛ X ✉♥❛ ✈✳❛✳r✳ ❝♦♥ ❞✐str✐❜✉❝✐ó♥ ✉♥✐❢♦r♠❡ ❡♥ [−1, 1]✳ ❉❡t❡r♠✐♥❛r 2 ❧❛s ❞✐str✐❜✉❝✐♦♥❡s ❞❡ |X| ② ❞❡ X ✳ EX U S LE A U N A M ❉❡✜♥✐❝✐ó♥ ❞❡ ❊s♣❡r❛♥③❛ ❈♦♥❞✐❝✐♦♥❛❧ ❊st❡ ❝❛♣ít✉❧♦ ♣r❡t❡♥❞❡ ❤❛❝❡r ✉♥ ❡st✉❞✐♦ s✐st❡♠át✐❝♦ ② r✐❣✉r♦s♦ ❞❡ ❧❛ ♥♦❝✐ó♥ ♣r♦❜❛✲❜✐❧íst✐❝❛ ❞❡ ❞❡♣❡♥❞❡♥❝✐❛ s✐❣✉✐❡♥❞♦ ❧❛s ♣❛✉t❛s ♠❛r❝❛❞❛s ♣♦r ❑♦❧♠♦❣♦r♦✈ ❡♥ ❧♦s ❛ñ♦s tr❡✐♥t❛✳ ❊♥ ❧♦ q✉❡ s✐❣✉❡ (Ω, A, P ) s❡rá ✉♥ ❡s♣❛❝✐♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞✳ ❙✐ A, B ∈ A ② P (A) > 0✱ s❡ ❞❡✜♥❡ ❧❛ ♣r♦❜❛❜✐❧✐❞❛❞ ❝♦♥❞✐❝✐♦♥❛❧ P (B|A) ❝♦♠♦ P (A∩B)/P (A)✳ ❙❡ ♦❜t✐❡♥❡ ❛sí ✉♥❛(B) = P (B A A ♣r♦❜❛❜✐❧✐❞❛❞ P ❡♥ (Ω, A) ❞❡✜♥✐❞❛ ♣♦r P |A)✱ B ∈ A✱ t♦t❛❧♠❡♥t❡ ❝♦♥❝❡♥✲(A) = 1 A tr❛❞❛ ❡♥ A ✭❡s ❞❡❝✐r✱ P ✮ ② q✉❡ ♥♦ ❡s ♠ás q✉❡ ❧❛ ♣r♦❜❛❜✐❧✐❞❛❞ P r❡str✐♥❣✐❞❛ −1 ❛❧ s✉❜❡s♣❛❝✐♦ ♠❡❞✐❜❧❡ ✭♥♦r♠❛❧✐③❛❞❛ ♣♦r ❡❧ ❢❛❝t♦r P (A) ♣❛r❛ q✉❡ P A s❡❛ ❡❢❡❝t✐✈❛✲♠❡♥t❡ ✉♥❛ ♣r♦❜❛❜✐❧✐❞❛❞✮✳ ❙✐ Y ❡s ✉♥❛ ✈✳❛✳r✳ ♥♦ ♥❡❣❛t✐✈❛ ♦ P ✲✐♥t❡❣r❛❜❧❡ ❡♥ (Ω, A)✱ s❡♣✉❡❞❡ ❞❡✜♥✐r ❧❛ ❡s♣❡r❛♥③❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ Y r❡s♣❡❝t♦ ❛❧ s✉❝❡s♦ A ✭✈❡r ❡❧ Pr♦❜❧❡♠❛ R −1 = P (A) Y dP ✹✳✻ ✮✱ q✉❡ ❞❡♥♦t❛r❡♠♦s E(Y |A)✱ ♠❡❞✐❛♥t❡ E(Y |A) = R Y dP A ✱ ❞❡A B t❛❧ s✉❡rt❡ q✉❡ P (B|A) = E(I |A)❀ s✐ ♣♦❞❡♠♦s r❡❛❧✐③❛r ✉♥ ❡①♣❡r✐♠❡♥t♦ ❛❧❡❛t♦r✐♦ ❞❡❧ q✉❡ s❡ ♦❜t✐❡♥❡♥ ♥ú♠❡r♦s r❡❛❧❡s ❞❡ ❛❝✉❡r❞♦ ❝♦♥ ❧❛ ❞✐str✐❜✉❝✐ó♥ ❞❡ Y ✱ ❧❛ ❧❡② ❢✉❡rt❡❞❡ ❧♦s ❣r❛♥❞❡s ♥ú♠❡r♦s ♥♦s ♣❡r♠✐t❡ ❛♣r♦①✐♠❛r E(Y |A) ♣♦r ❡❧ ✈❛❧♦r ♠❡❞✐♦ ❞❡ Y ❡♥ EX U ✉♥❛ s✉❝❡s✐ó♥ ❞❡ ♣r✉❡❜❛s ✐♥❞❡♣❡♥❞✐❡♥t❡s ❞❡❧ ❡①♣❡r✐♠❡♥t♦ s✐ só❧♦ t❡♥❡♠♦s ❡♥ ❝✉❡♥t❛ S LE ❡❧ ✈❛❧♦r ❞❡ Y ❞❡s♣✉és ❞❡ ❝♦♥♦❝❡r q✉❡ ❤❛ ♦❝✉rr✐❞♦ ❡❧ s✉❝❡s♦ A✳ ❊st❛s ❞❡✜♥✐❝✐♦♥❡s ❞❡ A U N ♣r♦❜❛❜✐❧✐❞❛❞ ❝♦♥❞✐❝✐♦♥❛❧ ② ❞❡ ❡s♣❡r❛♥③❛ ❝♦♥❞✐❝✐♦♥❛❧ r❡s♣❡❝t♦ ❛ ✉♥ s✉❝❡s♦ ❞❡ ♣r♦❜❛❜✐✲ A M ❧✐❞❛❞ ♥♦ ♥✉❧❛ ♥♦s ♣❡r♠✐t❡♥ ❞❡t❡❝t❛r ❧❛ ♣♦s✐❜❧❡ ❞❡♣❡♥❞❡♥❝✐❛ ❡♥tr❡ ❞♦s s✉❝❡s♦s ♦ ❡♥tr❡ ✶✹✹ ❈❆P❮❚❯▲❖ ✶✷✳ ❉❊❋■◆■❈■Ó◆ ❉❊ ❊❙P❊❘❆◆❩❆ ❈❖◆❉■❈■❖◆❆▲ AGUSTÍN GARCÍA NOGALES TEORÍAS DE LA MEDIDA Y DE LA PROBABILIDAD ✉♥ s✉❝❡s♦ ② ✉♥❛ ✈✳❛✳◆♦s ♣r♦♣♦♥❡♠♦s ❡♥ ❡st❛ s❡❝❝✐ó♥ ❡st✉❞✐❛r ❧❛ ❞❡♣❡♥❞❡♥❝✐❛ ❡♥tr❡ ♦tr♦ t✐♣♦ ❞❡ ♦❜❥❡✲ t♦s ♣r♦❜❛❜✐❧íst✐❝♦s✱ ❝♦♠♦ ♣✉❡❞❡♥ s❡r ❞♦s ✈✳❛✳ ♦ ✉♥❛ ✈✳❛✳ ② ✉♥❛ σ✲á❧❣❡❜r❛❀ ♣❛r❛ ❡❧❧♦ ♥❡❝❡s✐t❛♠♦s ❡①t❡♥❞❡r ❧❛s ❞❡✜♥✐❝✐♦♥❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞ ② ❡s♣❡r❛♥③❛ ❝♦♥❞✐❝✐♦♥❛❧❡s ❛ ✉♥❝♦♥t❡①t♦ ♠ás ❛♠♣❧✐♦✳ ❊❧ t❡♦r❡♠❛ s✐❣✉✐❡♥t❡✱ q✉❡ ❡s ❝♦♥s❡❝✉❡♥❝✐❛ ✐♥♠❡❞✐❛t❛ ❞❡❧ t❡♦r❡♠❛ ❞❡ ❘❛❞♦♥✲◆✐✲❦♦❞②♠ ✭❚❡♦r❡♠❛ ✶✶✳✷ ✮ ✱ ♥♦s ♣❡r♠✐t✐rá ♦❜t❡♥❡r ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡s❡❛❞❛ ❞❡ ❡s♣❡r❛♥③❛❝♦♥❞✐❝✐♦♥❛❧✳ ′ ′ , )❊♥ ❡❧ r❡st♦ ❞❡ ❡st❛ s❡❝❝✐ó♥✱ ♠✐❡♥tr❛s ♥♦ s❡ ❞✐❣❛ ❧♦ ❝♦♥tr❛r✐♦✱ (Ω A s❡rá ✉♥ ′ ′ , )❡s♣❛❝✐♦ ♠❡❞✐❜❧❡✱ X : (Ω, A) −→ (Ω A ✉♥❛ ✈✳❛✳ ❡ Y s❡rá ✉♥❛ ✈✳❛✳r✳ ❞❡✜♥✐❞❛ ❡♥(Ω, A) ♥♦ ♥❡❣❛t✐✈❛ ♦ ❝♦♥ ❡s♣❡r❛♥③❛ ✜♥✐t❛✳ ❊❖❘❊▼❆❚ ✶✷✳✶✳ ❊♥ ❧❛s ❝♦♥❞✐❝✐♦♥❡s ♣r❡❝❡❞❡♥t❡s✱ s✐ Y ❡s ♥♦ ♥❡❣❛t✐✈❛ ✭r❡s♣✳✱ s✐ Y ′ ′ , ) R t✐❡♥❡ ❡s♣❡r❛♥③❛ ✜♥✐t❛✮✱ ❡①✐st❡ ✉♥❛ ✈✳❛✳ g : (Ω A −→ ¯ ♥♦ ♥❡❣❛t✐✈❛ ✭r❡s♣✳✱ ❝♦♥ X ′ ′ ❡s♣❡r❛♥③❛ ✜♥✐t❛ r❡s♣❡❝t♦ ❛ P ✮ q✉❡✱ ♣❛r❛ ❝❛❞❛ A ✱ ✈❡r✐✜❝❛ q✉❡∈ A Z Z X Y (ω)dP (ω) = g(x)dP (x). ✭✶✷✳✶✮ − 1 ′ ′X (A ) A ❆❞❡♠ás✱ g ❡s ❡s❡♥❝✐❛❧♠❡♥t❡ ú♥✐❝❛✱ ❡♥ ❡❧ s❡♥t✐❞♦ ❞❡ q✉❡ ❝✉❛❧q✉✐❡r ♦tr❛ ✈✳❛✳ q✉❡ s❛t✐s❢❛❣❛ X ❧♦ ♠✐s♠♦ q✉❡ g ❝♦✐♥❝✐❞❡ ❝♦♥ g✱ P ✲❝✳s✳ P♦r ú❧t✐♠♦✱ s✐ Y t✐❡♥❡ ❡s♣❡r❛♥③❛ ✜♥✐t❛✱ ❡♥t♦♥❝❡s X g❡s ✜♥✐t❛ P ✲❝✳s✳ ❉❡♠♦str❛❝✐ó♥✳ ❙✉♣♦♥❣❛♠♦s ❡♥ ♣r✐♠❡r ❧✉❣❛r q✉❡ Y ❡s ♥♦ ♥❡❣❛t✐✈❛✳ ▲❛ ❢✉♥❝✐ó♥ ❞❡❝♦♥❥✉♥t♦s Z ′ ′ ′EX λ : A ) = Y (ω)dP (ω)∈ A −→ λ(A ∈ [0, +∞] X (A )S LE A 1 U− ′ ❡stá ❜✐❡♥ ❞❡✜♥✐❞❛ ✭♣♦r s❡r Y ♠❡❞✐❜❧❡ ♥♦ ♥❡❣❛t✐✈❛✮✱ ② ❞❡✜♥❡ ✉♥❛ ♠❡❞✐❞❛ ❡♥ ❧❛ σ✲ U′ ′ ′ −1 ′ N ) n( n A ) = á❧❣❡❜r❛ A ❀ ❡♥ ❡❢❡❝t♦✱ s✐ (A ❡s ✉♥❛ s✉❝❡s✐ó♥ ❞✐s❥✉♥t❛ ❡♥ A ✱ ❡♥t♦♥❝❡s X ∪ nn A M−1 ′ −1 ′ X (A ) (A )) n n ∪ n ② (X n ❡s ✉♥❛ s✉❝❡s✐ó♥ ❞✐s❥✉♥t❛ ❡♥ A❀ s❡ s✐❣✉❡ ❡♥t♦♥❝❡s ❞❡❧ t❡♦r❡♠❛