Giới thiệu về tập ngẫu nhiên

➤➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❤➭ ♥é✐ tr➢ê♥❣ ➤➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥ ❚r➬♥ ❚❤Þ ❍➯✐ ▲ý ●✐í✐ t❤✐Ư✉ ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ❦❤♦❛ ❤ä❝ ❍➭ ◆é✐ ✲ ✷✵✶✹ ➤➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❤➭ ♥é✐ tr➢ê♥❣ ➤➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥ ❚r➬♥ ❚❤Þ ❍➯✐ ▲ý ●✐í✐ t❤✐Ư✉ ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❈❤✉②➟♥ ♥❣➭♥❤✿ ▲ý t❤✉②Õt ①➳❝ s✉✃t ✈➭ t❤è♥❣ ❦➟ t♦➳♥ ❤ä❝ ▼➲ sè✿ ✻✵✳✹✻✳✶✺ ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ❦❤♦❛ ❤ä❝ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ♥❤ ❍➭ ◆é✐ ✲ ✷✵✶✹ ▼ơ❝ ❧ơ❝ ▼ë ➤➬✉ ✶ ✶✳ ❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✸ ✶✳✶✳ ❑✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò ①➳❝ s✉✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ▼ét ✈➭✐ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ t❤è♥❣ ❦➟ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✶✳ ▼✐Ò♥ t✐♥ ❝❐② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✷✳ ❚❤è♥❣ ❦➟ ❇❛②❡s ♠➵♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✸✳ P❤➞♥ tÝ❝❤ ❞÷ ❧✐Ö✉ t❤➠ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✹✳ ❚❤➠♥❣ t✐♥ ❞ù❛ tr➟♥ ♥❤❐♥ t❤ø❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✺✳ ▲✃② ♠➱✉ ①➳❝ s✉✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳ ❈➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❤÷✉ ❤➵♥ ✶✹ ✷✳✶✳ ❚❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♣❤➞♥ ❜è ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳ ❈➳❝ q✉❛♥ s➳t ❝ã ❣✐➳ trÞ t❐♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸✳ ❳➳❝ s✉✃t ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺✳ ❚❐♣ ➤ã♥❣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t➠♣➠ ❧✐➟♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✹✳ P❤➞♥ ❜è ❡♥tr♦♣② ❝ù❝ ➤➵✐ ✸✳ ❚❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➳❝ ✈✃♥ ➤Ò ❧✐➟♥ q✉❛♥ ✶ ✹✼ ✸✳✶✳ ▼è✐ q✉❛♥ ❤Ư ✶✲✶ ❣✐÷❛ ❤➭♠ ♣❤➞♥ ❜è ✈➭ ❤➭♠ ♠❐t ➤é ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷✳ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✸✳ ➜➵♦ ❤➭♠ ❘❛❞♦♥ ✲ ◆✐❦♦❞②♠ ✻✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑Õt ❧✉❐♥ ✼✶ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✼✶ ✷ ▼ë ➤➬✉ ▲ý t❤✉②Õt t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❧➭ t➢➡♥❣ ➤è✐ ♠í✐✳ ❈❤♦q✉❡t ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ✈➭✐ ý t➢ë♥❣ t❤❡♥ ❝❤èt ✈Ò t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭♦ ♥➝♠ ✶✾✺✸✱ ❑❡♥❞❛❧❧ ♥➝♠ ✶✾✼✹ ✈➭ ▼❛t❤❡r♦♥ ➤➲ ❝✉♥❣ ❝✃♣ ♥❤÷♥❣ ❝➡ së ♥Ị♥ ♠ã♥❣ ❝❤♦ ❧ý t❤✉②Õt ♥➭② ✈➭♦ ♥➝♠ ✶✾✼✺✳ ❚➭✐ ❧✐Ö✉ ✈Ị ❧ý t❤✉②Õt t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣ ❞➬♥ ❞➬♥ trë ♥➟♥ ❝ã ý ♥❣❤Ü❛ ❦Ó tõ ➤ã✳ ▼➷❝ ❞ï ❝ã ♥❤÷♥❣ ❦❤ã ❦❤➝♥ tr➢í❝ ♠➽t✱ ❦❤➠♥❣ ❝❤Ø ì tí ứ t ủ tí ợ ị trị t ò ì tế tố ♠➠ ❤×♥❤ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❞Ơ ①ư ❧ý✱ t✉② ♥❤✐➟♥ ❦❤➠♥❣ ♣❤➯✐ ✈× t❤Õ ♠➭ ❧ý t❤✉②Õt ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➤➢ỵ❝ ♥❤✐Ị✉ t➳❝ ❣✐➯ q✉❛♥ t➞♠✳ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ♠✉è♥ ❣✐í✐ t❤✐Ư✉ tỉ♥❣ q✉➳t ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♥❣❤✐➟♥ ❝ø✉ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥✳ ❉➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ♥❤✱ ❧✉❐♥ ✈➝♥ ❝đ❛ t➠✐ ❣å♠ ✸ ❝❤➢➡♥❣✿ ❈❤➢➡♥❣ ✶✿ ❚r×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❧ý t❤✉②Õt ①➳❝ s✉✃t ♠➭ t❛ ❝➬♥ ➤Õ♥ ➤Ó t❤➯♦ ❧✉❐♥ ✈Ò t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣ s❛✉✳ ❈❤Ø r❛ sù tå♥ t➵✐ ❝ñ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✱ ➤➷❝ ❜✐Öt ❧➭ tr♦♥❣ ❧ý t❤✉②Õt t❤è♥❣ ❦➟ t♦➳♥ ❤ä❝✳ ❈❤➢➡♥❣ ✷✿ ◆❣❤✐➟♥ ❝ø✉ tr➢ê♥❣ ❤ỵ♣ ❝đ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr ữ r ì ❈❆❘ ❝❤♦ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❞ù❛ ✈➭♦ ❧ý t❤✉②Õt ✈Ò t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ➜Ò ❝❐♣ ➤Õ♥ ❜➭✐ t♦➳♥ ❡♥tr♦♣② ❝ù❝ ➤➵✐ ❧✐➟♥ q✉❛♥ ❝❤♦ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❚r×♥❤ ❜➭② ✈Ị t❐♣ ➤ã♥❣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t➠♣➠ t❤Ý❝❤ ❤ỵ♣ ❝❤♦ ❧í♣ ❝➳❝ t❐♣ ❝♦♥ ➤ã♥❣ ❝đ❛ Rd ✳ ❈❤➢➡♥❣ ✸✿ ❈❤Ø r❛ ♠è✐ q✉❛♥ ❤Ư ✶✲✶ ❣✐÷❛ ❤➭♠ ♣❤➞♥ ❜è ✈➭ ❤➭♠ ♠❐t ➤é ❝ñ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❚r×♥❤ ❜➭② tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ✈➭ ➤➵♦ ❤➭♠ ❘❛❞♦♥ ✲ ◆✐❦♦❞②♠ ❝ñ❛ ❝➳❝ ❤➭♠ t❐♣ ❦❤➠♥❣ ❝é♥❣ tÝ♥❤✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ❚❙✳ ◆❣✉②Ơ♥ ị tỏ ò ết s s ✈➭ ❝❤➞♥ t❤➭♥❤ tí✐ t❤➬②✱ t❤➬② ➤➲ ✶ ❣✐❛♦ ➤Ị t ó ữ ị ú t tr♦♥❣ q✉➳ tr×♥❤ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ❈➯♠ ➡♥ ❝➳❝ t❤➬②✱ ❝➠ tr♦♥❣ ❑❤♦❛ ❚♦➳♥ ✲ ❈➡ ✲ ❚✐♥ ❤ä❝✱ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❑❤♦❛ ❤ä❝ ❚ù ♥❤✐➟♥ ✲ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐ ➤➲ ➤é♥❣ ✈✐➟♥✱ q✉❛♥ t➞♠ ✈➭ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ t➵✐ tr➢ê♥❣✳ ❈✉è✐ ❝ï♥❣✱ t➠✐ r✃t ♠♦♥❣ ♥❤❐♥ ➤➢ỵ❝ ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ q✉ý ❜➳✉ ❝đ❛ ❝➳❝ t❤➬②✱ ❝➠ ❣✐➳♦ ✈➭ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ➤Ó ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥✦ ❍➭ ◆é✐✱ ♥❣➭② ✵✶ t❤➳♥❣ ✵✶ ♥➝♠ ✷✵✶✹ ❚➳❝ ❣✐➯ ❚r➬♥ ❚❤Þ ❍➯✐ ▲ý ✷ ❈❤➢➡♥❣ ✶✳ ❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✶✳✶✳ ❑✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò ①➳❝ s✉✃t P❤➬♥ ♥➭② tr×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ò ❧ý t❤✉②Õt ①➳❝ s✉✃t ♠➭ t❛ ❝➬♥ ➤Õ♥ ➤Ĩ t❤➯♦ ❧✉❐♥ ✈Ị ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ ❝➳❝ s ị ĩ ì t ọ ệ tợ ì t ọ ♠ét ♣❤Ð♣ t❤ö ♥❣➱✉ ♥❤✐➟♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✭Ω ✱ A✱ P ✮✱ tr♦♥❣ ➤ã✿ α✮ Ω ❧➭ ♠ét t❐♣✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❦❤➠♥❣ ❣✐❛♥ ♠➱✉ ❝đ❛ ♣❤Ð♣ t❤ư✳ β ✮ A ❧➭ ♠ét σ ✲➤➵✐ sè ✭❜✐Ĩ✉ ❞✐Ơ♥ ❝➳❝ ❜✐Õ♥ ❝è✮✱ tø❝ ❧➭ ✿ ✭✐✮ Ω ∈ A ✭✐✐✮ ◆Õ✉ A ∈ A t❤× Ac ∈ A ✭✐✐✐✮ ◆Õ✉ An ∈ A ✈í✐ n ≥ t❤× An ∈ A✳ n≥1 ❈➷♣ (Ω, A) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤♦ ➤➢ỵ❝✳ γ ✮ P : A → [0, 1] ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é ➤♦ ①➳❝ s✉✃t✱ tø❝ ❧➭ ✿ ✐✮ P (Ω) = ✸ ✐✐✮ ◆Õ✉ {An , n ≥ 1} ❧➭ ♠ét ❞➲② ✭❤÷✉ ❤➵♥ ❤♦➷❝ ✈➠ ❤➵♥ ➤Õ♠ ➤➢ỵ❝✮ ❝đ❛ ❝➳❝ ♣❤➬♥ tư rê✐ ♥❤❛✉ tõ♥❣ ➤➠✐ ♠ét ✭Ai ∩Aj = ∅ ✈í✐ i=j ✮ t❤× P( An ) = n≥1 P (An ) n≥1 ❚Ý♥❤ ❝❤✃t ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ σ ✲ ❝é♥❣ tÝ♥❤ ❝đ❛ P ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷✳ ✭P❤➬♥ tư ♥❣➱✉ ♥❤✐➟♥✮ ❈❤♦ ✭Ω ✱ A✱ P ✮ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳ ▼ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X ❧➭ ♠ét ➳♥❤ ①➵ tõ Ω tí✐ R s❛♦ ❝❤♦ X −1 (B(R)) ⊆ A✱ tø❝ ❧➭✱ ∀B ∈ B(R)✱ X −1 (B) ∈ A✱ ♥ã✐ ❝➳❝❤ ❦❤➳❝✱ X ❧➭ ♠ét ➳♥❤ ①➵ A ✲ B(R) ✲ ➤♦ ➤➢ỵ❝✳ ❚r♦♥❣ ➤ã B(R) ❧➭ σ ✲ tr➢ê♥❣ ❇♦r❡❧ ➤➢ỵ❝ s✐♥❤ r❛ ❜ë✐ ❝➳❝ t❐♣ ♠ë ❝ñ❛ R✱ X −1 (B) = {ω : X(ω) ∈ B}✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✸✳ ✭▲✉❐t ①➳❝ s✉✃t ❝đ❛ ❝➳❝ ♣❤➬♥ tö ♥❣➱✉ ♥❤✐➟♥✮ ❈❤♦ ✭Ω ✱ A✱ P ✮ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈➭ (U, U) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤♦ ➤➢ỵ❝✳ ➪♥❤ ①➵ X : Ω → U ❧➭ A ✲ U ✲ ➤♦ ➤➢ỵ❝✳ ▲✉❐t ①➳❝ s✉✃t ❝ñ❛ X ❧➭ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥ U ợ ị ĩ PX = P X ị ♥❣❤Ü❛ ✶✳✶✳✹✳ ✭❍➭♠ ♣❤➞♥ ❜è ❝ñ❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✮ ❈❤♦ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X : (Ω, A, P ) → (R, B(R)) ✈➭ ❧✉❐t ①➳❝ s✉✃t ❝ñ❛ ♥ã✿ PX = P X −1 tr➟♥ B(R)✳ ❍➭♠ ♣❤➞♥ ❜è ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X ❧➭ ❤➭♠ F : R [0, 1] ợ ị ĩ F (x) = PX ((−∞, x]) ❚Ý♥❤ ❝❤✃t ✶✳ ❍➭♠ F ♥➭② t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ s❛✉✿ ✭✐✮ F ❧➭ ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ ❣✐➯♠✱ tø❝ ❧➭✱ ♥Õ✉ x ≤ y t❤× F (x) ≤ F (y)✱ ✭✐✐✮ lim F (x) = ❀ lim F (x) = x +∞ x −∞ ✭✐✐✐✮ F ❧➭ ❧✐➟♥ tô❝ ♣❤➯✐ tr➟♥ R✱ tứ ỗ x R F (x) = lim F (y) = y + F (x )✱ ✈➭ ❝ã ❣✐í✐ ❤➵♥ tr➳✐ t➵✐ ♠ä✐ x ∈ R✳ ✹ x ❚✃t ❝➯ ❝➳❝ ❤➭♠ tr➟♥ R t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ✭✐✮✱ ✭✐✐✮✱ ✭✐✐✐✮ ë tr➟♥ ❧➭ ❝➳❝ ❤➭♠ ♣❤➞♥ ❜è ❝ñ❛ ❝➳❝ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥ B(R)✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❝ã ♠ét s♦♥❣ ➳♥❤ ❣✐÷❛ ❝➳❝ ❤➭♠ t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ✭✐✮✱ ✭✐✐✮✱ ✭✐✐✐✮ ë tr➟♥ ✈í✐ ❝➳❝ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳ B(R)✳ ✭❍➭♠ ♣❤➞♥ ❜è ❝ñ❛ ✈Ð❝ t➡ ♥❣➱✉ ♥❤✐➟♥✮ ❈❤♦ X : (Ω, A, P ) → (Rd , B(Rd )) ✭X = (X1 , · · · , Xd ) ❧➭ ✈Ð❝ t➡ ♥❣➱✉ ♥❤✐➟♥ d ❝❤✐Ò✉✮✳ ❍➭♠ ♣❤➞♥ ❜è F ❝ñ❛ X ❧➭ ❤➭♠✿ F : Rd → [0, 1] F (x) = P (X ≤ x) = P (X1 ≤ x1 , · · · , Xd ≤ xd ) = P X −1 ((−∞, x]), ∀x = (x1 , · · · , xd ) ∈ Rd ❚Ý♥❤ ❝❤✃t ✷✳ ❚õ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ P, F t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ✭✐✮ ≤ F (x) ≤ 1✱ ✭✐✐✮ lim F (x1 , · · · , xd ) = ✈í✐ Ýt ♥❤✃t ♠ét ❥ ♥➭♦ ➤ã✱ ✈➭ xj →−∞ lim F (x1 , · · · , xd ) = ✈í✐ t✃t ❝➯ j = 1, 2, · · · , d✳ xj →+∞ ✭✐✐✐✮ F ❧➭ ❧✐➟♥ tô❝ ♣❤➯✐ tr➟♥ Rd ✱ tø❝ ❧➭ lim F (y) = F (x), ∀x ∈ Rd y ✶✳✷✳ x ▼ét ✈➭✐ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ t❤è♥❣ ❦➟ P❤➬♥ ♥➭② ❝❤Ø r❛ sù tå♥ t➵✐ ❝ñ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✱ ➤➷❝ ❜✐Öt ❧➭ tr♦♥❣ ❧ý t❤✉②Õt t❤è♥❣ ❦➟ t♦➳♥ ❤ä❝✳ ✶✳✷✳✶✳ ▼✐Ị♥ t✐♥ ❝❐② ❳Ðt ♠ét ♠➠ ❤×♥❤ t❤è♥❣ ❦➟ t❤❛♠ sè ❤ã❛✳ {f (x, θ) : x ∈ X ⊆ Rm , θ ∈ ϕ(θ)✳ X ⊆ Rd }✱ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ❤➭♠ ♠❐t ➤é X1 , X2 , · · · , Xn ♥❣➱✉ ♥❤✐➟♥ ❝ñ❛ ✈➭ ♠ét t❤❛♠ sè ♠➭ t❛ q✉❛♥ t➞♠ f (x, θ)✱ ❝❤♦ tr➢í❝ ♠ét ♠➱✉ ♥❣➱✉ ♥❤✐➟♥ X ✱ ❜➯♥ ❝❤✃t ủ ợ ề t ó tì ♠ét t❐♣ C(X1 , X2 , · · · , Xn ) ♠➭ ❝❤ø❛ ϕ(θ0 )✱ θ0 ❧í♥✳ ❈ơ t❤Ĩ✱ t❐♣ ♥❣➱✉ ♥❤✐➟♥ C(X1 , X2 , · · · ✺ ❧➭ t❤❛♠ sè t❤ù❝✱ ✈í✐ ①➳❝ s✉✃t , Xn ) ❧➭ ♠ét t❐♣ t✐♥ ❝❐② ❝❤♦ ϕ(θ) ✈í✐ ♠ø❝ t✐♥ ❝❐② − α ∈ (0, 1) ♥Õ✉ ∀θ ∈ ❚r♦♥❣ ➤ã : Pθ (ϕ(θ) ∈ C(X1 , X2 , · · · , Xn )) ≥ − α✳ dPθ = f (x, θ)dx✳ ❚r♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ➤➡♥ ❣✐➯♥✱ ✈✐Ư❝ ①➞② ❞ù♥❣ ♠✐Ị♥ t✐♥ ❝❐② tèt ♥❤✃t ❝❤♦ ϕ(θ) ❝ã t❤Ĩ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ ♠➭ ❦❤➠♥❣ ❝➬♥ sư ❞ơ♥❣ ❦❤➳✐ ♥✐Ư♠ ❤×♥❤ t❤ø❝ ✈Ị ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➳❝ ♣❤➞♥ ❜è ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❱Ý ❞ô✿ ❈❤♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X ❝ã ♣❤➞♥ ❜è ❝❤✉➮♥ N (µ, σ )✱ ë ➤➞② θ = (µ, σ )✳ √ n(Xn − µ)/V ❝ã ♣❤➞♥ ♣❤è✐ ❙t✉❞❡♥t ✈í✐ n − ❜❐❝ tù n X + · · · + X 2 n ❞♦✱ tr♦♥❣ ➤ã Xn = ✈➭ V = n n − i=1(Xi − Xn ) ❧➭ tr✉♥❣ ❜×♥❤ ♠➱✉ ✈➭ ♣❤➢➡♥❣ s❛✐ ♠➱✉ ❤✐Ö✉ ❝❤Ø♥❤✱ ❞ù❛ tr➟♥ ♠ét ♠➱✉ ♥❣➱✉ ♥❤✐➟♥ X1 , · · · , Xn ➤➢ỵ❝ ❳Ðt ϕ(θ) = µ✳ ❑❤✐ ➤ã X ✳ ❉♦ ➤ã✱ ♠ét ❦❤♦➯♥❣ t✐♥ ❝❐② (1 − √α) ✪ ❝❤♦ µ ❝ã t❤Ĩ t❤✉ ➤➢ỵ❝ ♥❤➢ s❛✉✿ n n t1 < V (Xn − µ) < t2 s❛♦ ❝❤♦ P (t1 < V (Xn − µ) < t2 ) = − α✳ ⇒ L(X1 , · · · , Xn ) < µ < U (X1 , · · · , Xn ) rót r❛ √ tõ ❚r♦♥❣ ➤ã✿ L = Xn − V √ t ❀ n U = Xn − V √ t n √ n (t1 , t2 ) s❛♦ ❝❤♦✿ P (t1 < V (Xn − µ) < t2 ) = − α✳ ❑❤♦➯♥❣ t✐♥ ❝❐② tèt ♥❤✃t t➵✐ ♠ø❝ − α ó tể ợ ị ĩ ể ❝ã ♥❤✐Ị✉ ➤✐Ĩ♠ t✐♥ ❝❐② ✈í✐ ➤é ❞➭✐ ♥❤á ♥❤✃t ✭❜✐Ĩ✉ ❞✐Ơ♥ ➤é ❝❤Ý♥❤ ①➳❝ ❝đ❛ ➢í❝ ❧➢ỵ♥❣ ❝❤♦ ϕ(θ)✮✳ ➜é ❞➭✐ V (t − t )✱ ♥➟♥ |S| = U − L = √ n ❝❤ó♥❣ t❛ ❝ã t❤Ó ❝❤ä♥ t1 , t2 s❛♦ ❝❤♦ ❝ù❝ t✐Ó✉ ➤é ❞➭✐ ❦× ✈ä♥❣ E|S| ❝đ❛ t❐♣ ❝♦♥ ♥❣➱✉ ë ➤➞② ❝❤Ý♥❤ ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❝ơ t❤Ĩ ❧➭✱ ♥❤✐➟♥ S = [L, U ]✳ ❚r♦♥❣ ❝➳❝ ❝❤✐Ị✉ ❧í♥ ❤➡♥✱ ➤é ❞➭✐ ❝ñ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ S t❤❛② t❤Õ ❜ë✐ ❦❤è✐ ▲❡❜❡s❣✉❡ ∧(S)✱ ♥➟♥ t❛ sÏ ♣❤➯✐ tÝ♥❤ t♦➳♥ E∧(S)✳ ➤✐Ị✉ ♥➭②✱ t❛ ❝➬♥ ➤Þ♥❤ ♥❣❤Ü❛ ❦❤➳✐ ♥✐Ư♠ ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ ➤➢ỵ❝ ➜Ĩ ❧➭♠ ➤➢ỵ❝ Rd ✱ t❤❡♦ ➤ã t❤× ∧(S) ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✭❦❤➠♥❣ ➞♠✮✳ ✶✳✷✳✷✳ ❚❤è♥❣ ❦➟ ❇❛②❡s ♠➵♥❤ ❚r➢í❝ ❦❤✐ ♥ã✐ ✈Ị ❤Ư ♣❤➢➡♥❣ ♣❤➳♣ ❇❛②❡s ♠➵♥❤✱ t❛ ➤➢❛ r❛ ♠ét ✈Ý ❞ơ ❝đ❛ tr➢ê♥❣ ❤ỵ♣ ✧❝➳❝ ①➳❝ s✉✃t ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝✧✳ ❱Ý ❞ô✿ ❈❤♦ X ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ {a, b, c, d} ⊆ R ✈í✐ ♠❐t ➤é f0 ❦❤➠♥❣ ❤♦➭♥ ❝❤Ø♥❤ ➤➢ỵ❝ ①➳❝ ị ợ ệ q ỉ rõ ố q✉❛♥ ❤Ư ❣✐÷❛ ❤➭♠ ♣❤➞♥ ❜è ✈➭ ❤➭♠ ♠❐t ➤é ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ ♥❤➢ s❛✉✿ ❍Ư q✉➯ ✸✳✶✳✸✳ ❈❤♦ D ❧➭ t❐♣ ❝➳❝ ♠❐t ➤é tr➟♥ 2U ✈í✐ ❣✐➳ trÞ t➵✐ ∅✱ ✈➭ B ❧➭ ξ t❐♣ ❝➳❝ ❤➭♠ ♣❤➞♥ ❜è tr➟♥ 2U ✳ ❑❤✐ ➤ã D → B ❧➭ t➢➡♥❣ ø♥❣ ✶✲✶ ✈í✐ ♥❣❤Þ❝❤ ➤➯♦ ❧➭ ứ ệ q ợ s r từ ị ❧ý ✸✳✶✳✶✺ ë tr➟♥✳ ❍Ö q✉➯ ✸✳✶✳✹✳ ▼ét ❤➭♠ ♣❤➞♥ ❜è F ❧➭ ♠ét ➤é ➤♦ ①➳❝ s✉✃t ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ F ∗ µ ❧➭ ♠ét ♠❐t ➤é tr➟♥ U ✱ tø❝ ❧➭✱ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ (F ∗ µ)(X) = ✈í✐ ♠ä✐ X ♠➭ |X| ≥ 2✳ ❈❤ø♥❣ ♠✐♥❤✳ x∈X ❍Ư q✉➯ ❞Ơ ❞➭♥❣ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ✈í✐ ❝❤ó ý r➺♥❣ f (x)✱ ✈í✐ f ✸✳✷✳ ❧➭ ♠❐t ➤é tr➟♥ U F ❝❤♦ ➤é ➤♦ tr➟♥ F (X) = 2U ✳ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t P❤➬♥ ♥➭② tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ tÝ❝❤ ♣❤➞♥ ❝đ❛ ❝➳❝ ❤➭♠ t❐♣ ❦❤➠♥❣ ❝é♥❣ tÝ♥❤✱ ♠➭ t❛ ❣ä✐ ❧➭ tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t✳ ❈❤♦ X ột ế ợ ị ♥❣❤Ü❛ tr➟♥ ❑❤✐ ➤ã✿ (Ω, A, P )✳ +∞ EX = X(ω)dP (ω) = P (X > t)dt Ω ❚❛ ➤➲ ①Ðt tr♦♥❣ ♣❤➬♥ ✷✳✸ r➺♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ t❤➠♥❣ t✐♥ t❤è♥❣ ❦➟ ❦❤➠♥❣ ➤➬② ➤đ✱ ❝❤➷♥ ❞➢í✐ ❝đ❛ ❝➳❝ ❣✐➳ trÞ ❦ú ✈ä♥❣ ❝đ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♣❤➞♥ ❜è F X ➤➢ỵ❝ ✈✐Õt t❤❡♦ ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❑❤✐ ➤ã✱ t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ ❦❤ë✐ ➤➬✉ ❝đ❛ ❈❤♦q✉❡t ✈Ị tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ♥❤➢ s❛✉✿ ✺✾ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✷✸✳ ❈❤♦ X ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠ ợ ị ĩ tr (, A, P ) F ♠ét ❤➭♠ t❐♣ ➤➡♥ ➤✐Ư✉ tr➟♥ A ✳ ❇✐Ĩ✉ t❤ø❝✿ +∞ F (X > t)dt ➤➢ỵ❝ ❣ä✐ ❧➭ tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❝đ❛ X ➤è✐ ✈í✐ ❤➭♠ t❐♣ ➤➡♥ ➤✐Ư✉ F ✭❦❤➠♥❣ ♥❤✃t t❤✐Õt ❝é♥❣ tÝ♥❤✮✳ ❚❛ sÏ t❤✐Õt ❧❐♣ ♠ét ❦❤➳✐ ♥✐Ư♠ ❦❤➳ tỉ♥❣ q✉➳t ✈Ị tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❝ñ❛ ❝➳❝ ❤➭♠ t❐♣ ♠➭ ❦❤➠♥❣ ❝➬♥ ❧➭ σ ✲ ❝é♥❣ tÝ♥❤✳ ❑❤➳✐ ♥✐Ư♠ ♥➭② ❝đ❛ tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t sÏ tỉ♥❣ q✉➳t ❤ã❛ tÝ❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡✳ ❈❤♦ ❧í♣ Ω ❧➭ ột t t trị tự ợ ị ĩ tr A ó t ủ ó ứ t rỗ s µ(∅) = 0✱ ✈➭ ∀A, B ∈ A ✈í✐ A ⊆ B ✱ t❛ ❝ã µ(A) ≤ µ(B)✱ ♥❣❤Ü❛ ❧➭✱ µ ❧➭ ♠ét ❤➭♠ t❐♣ ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ ❣✐➯♠✳ ❚❛ ①Ðt ✧tÝ❝❤ ♣❤➞♥✧ ❝đ❛ ❝➳❝ ❤➭♠ f ♥❤❐♥ ❣✐➳ trÞ tự ợ ị ĩ tr ể t tệ t ré♥❣ ❝➳❝ ❣✐➳ trÞ ❝ã t❤Ĩ tí✐ ❤➭♠ ❧➭ ➤♦➵♥ A ⊆ Ω}✱ tí✐ [−∞, +∞]✱ Ω✳ ±∞✱ ♥➟♥ ♠✐Ị♥ ❣✐➳ trÞ ❝đ❛ ❝➳❝ ✈➭ t✃t ❝➯ ❝➳❝ ❣✐➳ trÞ ❝ù❝ ➤➵✐✱ ♥❤➢ tå♥ t➵✐✳ ❈➳❝ ♣❤Ð♣ t♦➳♥ sè ❤ä❝ ➤➢ỵ❝ ♠ë ré♥❣ tõ sup{f (ω) : ω ∈ R = (−∞, +∞) R = [−∞, +∞] t❤❡♦ ❝➳❝❤ t❤➠♥❣ t❤➢ê♥❣✳ ●✐➯ sö B(R) A ❧➭ ♠ét σ ✲ ➤♦ ➤➢ỵ❝ ✈í✐ ✲ tr➢ê♥❣ ❝➳❝ t❐♣ ❝♦♥ ❝đ❛ B(R) ❧➭ σ Ω✳ ✲ tr➢ê♥❣ ❇♦r❡❧ tr➟♥ ❈❤♦ R f : Ω → R ➤➢ỵ❝ s✐♥❤ ❜ë✐ ❧➭ A− B(R) ✈➭ {−∞, +∞}✳ ❈❤ó ý✳ B(R+ ) = B([0, +∞]) = {A ∩ [0, +∞] : A ∈ B(R)} = {A : A ⊆ [0, +∞] ∩ B(R)}✳ ➜Ó tr➳♥❤ ❝➳❝ ❜✐Ó✉ t❤ø❝ ✈➠ ♥❣❤Ü❛ ♥❤➢ −∞, +∞✱ t❛ ①Ðt t➳❝❤ rê✐ ❝➳❝ ❤➭♠ ❦❤➠♥❣ ➞♠ ✈➭ ❦❤➠♥❣ ❞➢➡♥❣✳ ❈❤♦ f : Ω → R+ f1 , f2 , · · · ❧➭ A − B(R+ ) ❝ñ❛ ❝➳❝ ❤➭♠ ➤➡♥ ❣✐➯♥ ✲ ➤♦ ➤➢ỵ❝✱ ❦❤✐ ➤ã tå♥ t➵✐ ♠ét ❞➲② t➝♥❣ fn : Ω → [0, +∞]✱ ✻✵ ❧➭ ❝➳❝ ❤➭♠ ❝ã ❞➵♥❣ n aj 1Aj (ω) ✈í✐ aj ∈ R+ ✈➭ A1 , A2 , · · · , An ❧➭ ❝➳❝ ♣❤➬♥ tö rê✐ ♥❤❛✉ tõ♥❣ ➤➠✐ j=1 ❝ñ❛ A✱ s❛♦ ❝❤♦ ∀ω ∈ Ω, f (ω) = lim fn (ω)✳ n→∞ f : Ω → [−∞, +∞]✱ t❤× t❛ ✈✐Õt✿ ◆Õ✉ f (ω) = f + (ω) − f − (ω) ❱í✐ ❝➯ f + ✈➭ f − ❧➭ ❝➳❝ ➳♥❤ ①➵ tõ Ω f+ ❑❤✐ ✈➭ f− ❧➭ ➤♦ ➤➢ỵ❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❧➭ ➤♦ ợ ột ộ ủ ột f ✲ ❝é♥❣ tÝ♥❤ ❦❤➠♥❣ ➞♠ tr➟♥ ❦❤➠♥❣ ➞♠ ➤è✐ ✈í✐ f (ω)dµ(ω) = Ω ❱í✐ [0, +∞]✱ ✈➭ f tớ (, A) tí s ợ ị ĩ ❧➭✿ lim fn (ω)dµ(ω) = lim n→∞ Ω f1 , f2 , · · · fn (ω)dµ(ω) n→∞ Ω ❧➭ ❞➲② t➝♥❣ ❝➳❝ ❤➭♠ ➤➡♥ ❣✐➯♥ ✈➭ ❤é✐ tơ tí✐ f ✱ ✈➭ kn fn (ω)dµ(ω) = aj µ(Aj ) j=1 ♥Õ✉ ❱× A1 , A2 , · · · , Akn µ t➵♦ t❤➭♥❤ ♠ét ❧➭ ❝é♥❣ tÝ♥❤ ♥➟♥ ➤➵✐ ❧➢ỵ♥❣ A ✲ ♣❤➞♥ ❤♦➵❝❤ ❝đ❛ Ω ✳ f (ω)dµ(ω) Ω ❧❐♣ ✈í✐ ❝➳❝ ❝➳❝❤ ❝❤ä♥ r✐➟♥❣ ❜✐Ưt ❝đ❛ ❈❤♦ fn ✳ f : Ω → R ➤♦ ➤➢ỵ❝✱ t ị ĩ f + ()dà() < ∞ ✈í✐ f − (ω)dµ(ω) f + (ω)dµ(ω) − f (ω)dµ(ω) = f − (ω)dµ(ω) < ∞ ✈➭ Ω ❑❤✐ ợ ị ĩ tốt ó ộ à() < tì f ()dà() = à(f > t)dt + −∞ ✻✶ [µ(f > t) − µ(Ω)]dt (3.1) ❚❤❐t ✈❐② ✱ ✈× f + ≥ 0✱ t❛ ❝ã✿ ∞ f + (ω)dµ(ω) = ∞ µ(f + > t)dt = Ω µ(f > t)dt ∞ f − ()dà() = tự, ỗ t > 0, (f − > t) ❚❤❡♦ tÝ♥❤ ❝é♥❣ tÝ♥❤ ❝ñ❛ µ(f − > t)dt ✈➭ (f ≥ −t) t➵♦ t❤➭♥❤ ♠ét ♣❤➞♥ ❤♦➵❝❤ ❝đ❛ Ω✳ µ✱ µ(f − > t) = µ(Ω) − µ(f ≥ −t)✱ s✉② r❛ ✭✸✳✶✮ ❱➱♥ tr trờ ợ f dà = A à() < ❝❤♦ A ∈ A✱ t➢➡♥❣ tù t❛ ♥❤❐♥ ➤➢ỵ❝✿ (1A f )dµ Ω ∞ µ((f > t) ∩ A) − µ(A) dt µ((f > t) ∩ A)dt + = −∞ ∞ ❈❤♦ f ≥ 0✱ t❛ ❝ã f dµ = Ω sè t −→ µ(f > t) µ(f > t)dt✳ ❱Õ ♣❤➯✐ ❧➭ ♠ét tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ ➤è✐ ✈í✐ ➤é ➤♦ ▲❡❜❡s❣✉❡ dt tr➟♥ [0, ∞]✳ ó ợ ị ĩ tốt ễ (, A) t f f ợ ì ế A B[0, ] ợ tì (f > t) A t [0, ] ì t −→ µ(f > t) ❧➭ ❤➭♠ ❣✐➯♠ ✈➭ ♥ã ❧➭ B([0, ∞]) − B([0, ∞]) ✲ ➤♦ ➤➢ỵ❝✳ ➜✐Ị✉ ♥➭② ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ♥❤➢ s❛✉✿ ❈❤♦ ❑❤✐ ➤ã✿ t ∈ [0, ∞]✱ ➤➷t ϕ(t) = µ(f > t)✳  [a, ∞] ♥Õ✉ ∃a = inf{ϕ < s} (ϕ < s) = (a, ] ế inf{ < s} r ỗ trờ ợ t ế inf{ ột ❤➭♠ ➤➡♥ ➤✐Ö✉ tr➟♥ (ϕ < s) ∈ B([0, ∞])✳ < s} t ợ t a tì b a✱ t❛ ❝ã ϕ(b) ≤ ϕ(a) < s b ∈ ( < s) ợ ế (c) < s tì a ≤ c t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ a✳ ✻✷ ◆Õ✉ a = inf{ϕ < s} ❦❤➠♥❣ t❤✉ ➤➢ỵ❝✱ ❦❤✐ ➤ã ♥Õ✉ ϕ(b) < s✱ t❛ ❧✉➠♥ ❝ã a < b✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ ◆❤➢♥❣ y ❦❤✐ ➤ã tå♥ t➵✐ s❛♦ ❝❤♦ ϕ(y) < s ✈➭ a < y < c✳ ϕ ❧➭ ❣✐➯♠✱ ♥➟♥ ϕ(c) ≤ ϕ(y) < s✳ ❉♦ ➤ã c ∈ (ϕ < s)✳ ❱❐② ❝❤♦ ✈í✐ c > a✱ f : Ω → [0, ∞]✱ ∞ µ Cà (f ) = t ó tể ị ĩ tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❝đ❛ f ➤è✐ µ(f > t)dt✳ ❑❤✐ ➤ã✱ t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ tỉ♥❣ q✉➳t ❝đ❛ tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ♥❤➢ s❛✉✿ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✷✹✳ ❑❤✐ f : Ω → [−∞, +∞] ✈➭ µ(Ω) < ∞✱ tÝ❝❤ ♣❤➞♥ qt ủ f ố ợ ị ĩ ∞ Cµ (f ) = µ(f > t)dt + µ(f > t) − µ(Ω) dt −∞ ❚❛ ❝ò♥❣ ✈✐Õt Cµ (f ) = f dµ✳ ❱➭ ❝❤♦ A ∈ A✱ Ω ∞ Cµ (1A f ) = µ((f > t) ∩ A)dt + f dµ = ❚❛ ♥ã✐ r➺♥❣ f ❧➭ µ((f > t) ∩ A) − µ(A) dt −∞ A ❑❤✐ µ ✲ ❦❤➯ tÝ❝❤ tr➟♥ A ❦❤✐ Cµ (1A f ) < ∞✳ µ(Ω) < , à() = tì ế f ó ❜è✧  µ(f > t) ♥Õ✉ t ≥ ϕf (t) = µ(f ≥ t) − µ(Ω) ♥Õ✉ t < trị ì ọ ủ f t tÝ❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡ ❝ñ❛ ♣❤➞♥ ❜è ϕf (t)✱ ➤ã ❧➭✿ ∞ ϕf (t)dt −∞ ∞ (Ef = f dµ = Ω ❑❤✐ f = 1A ✈í✐ ∞ [µ(f ≥ t) − µ(Ω)]dt = µ(f > t)dt + −∞ ϕf (t)dt) −∞ A ∈ A✱ t❛ ❝ã µ(f > t) = µ(1A > t) = µ(A)1[0,1) (t) ✻✸ ∞ Cµ (1A ) = ❉♦ ➤ã µ(f > t)dt = n ❚æ♥❣ q✉➳t ❤➡♥✱ ❝❤♦ f (ω) = µ(A)dt = µ(A) 1Ai (ω) ✈í✐ A1 , · · · , An ❧➭ ❝➳❝ t❐♣ ❝♦♥ rê✐ i=1 ♥❤❛✉ tõ♥❣ ➤➠✐ ❝ñ❛ Ω a0 = < a1 < · · · < an (f ≥ 0)✱ t❛ ❝ã✿ ✈➭ n n µ(f > t) = µ i=1 ∞ n µ i=1 = n µ i=1 a i−1 ❚Ý♥❤ ❝❤✃t ✹✳ Aj 1[ai−1 ,ai ) (t)dt j=i n j=i n Cµ (f ) = ◆➟♥ Aj 1[ai−1 ,ai ) (t) n (ai − ai−1 )µ Aj dt = j=i n i=1 Aj j=i ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ✐✮ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❧➭ ➤➡♥ ➤✐Ö✉ ✈➭ ➤å♥❣ ♥❤✃t ❞➢➡♥❣ ❜❐❝ ✶ ✐✐✮ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❦❤➠♥❣ ❝é♥❣ tÝ♥❤✳ ❚Ý♥❤ ❝❤✃t ✐✮✿ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❧➭ ➤➡♥ ➤✐Ö✉ ✈➭ ➤å♥❣ ♥❤✃t ❞➢➡♥❣ ❜❐❝ ✶✱ ♥❣❤Ü❛ ❧➭✿ f ≤ g ⇒ Cµ (f ) ≤ Cµ (g) (f ≤ g ⇒ {f > t} ≤ {g > t} ⇒ µ(f > t) ≤ µ(g > t) ⇒ Cµ (f ) ≤ Cµ (g))✱ ✈➭ ❝❤♦ λ > ✱ Cµ (λf ) = λCµ (f )✳ ❚Ý♥❤ ❝❤✃t ✐✐✮✿ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ❦❤➠♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝é♥❣ tÝ♥❤✳ ❱Ý ❞ô✱ ♥Õ✉ f = (1/4)1A ✈➭ g = (1/2)1B ✈í✐ A ∩ B = ∅ t❤× f + g = (1/4)1A + (1/2)1B ✻✹ +∞ ⇒ Cµ (f + g) = µ(f + g > t)dt 1 1 = µ(A ∪ B) + ( − )µ(B) = µ(A ∪ B) + µ(B) 4 4 +∞ ▼➭ µ(f > t)dt = µ(A) Cµ (f ) = +∞ µ(g > t)dt = µ(B) Cµ (g) = ❙✉② r❛✿ ✭µ(A Cµ (f + g) = Cµ (f ) + Cµ (g) ∪ B) = µ(A) + µ(B) ❞♦ µ ❦❤➠♥❣ ♣❤➯✐ ❝é♥❣ tÝ♥❤✮ ❚✉② ♥❤✐➟♥✱ ♥Õ✉ t❛ ①Ðt ❤❛✐ ❤➭♠ ➤➡♥ ❣✐➯♥ f = a1A + b1B g = α1A + β1B ✈í✐ ✈í✐ f ✈➭ g ❝ã ❞➵♥❣ A ∩ B = ∅✱ ≤ a ≤ b✱ ≤ α ≤ β✳ ❚❤× Cµ (f + g) = Cµ (a1A + b1B + α1A + β1B ) = Cµ (a + α)1A + (b + β)1B = (a + α)µ(A ∪ B) + (b + β − a − α)µ(B) = aµ(A ∪ B) + (b − a)µ(B) + αµ(A ∪ B) + (β − α)µ(B) = Cµ (a1A + b1B ) + Cµ (α1A + β1B ) = Cµ (f ) + Cµ (g) n ❚ỉ♥❣ q✉➳t ❤➡♥✱ ➤➻♥❣ t❤ø❝ ♥➭② ➤ó♥❣ ❝❤♦ f= n aj 1Aj ✈➭ g= j=1 A1 , · · · , An ❧➭ rê✐ ♥❤❛✉ tõ♥❣ ➤➠✐✱ ✈➭ ≤ a1 ≤ a2 ≤ · · · ≤ b2 ≤ · · · ≤ bn ✳ ✻✺ bj 1Aj ✱ ✈í✐ j=1 an ✱ ≤ b1 ≤ ✸✳✸✳ ➜➵♦ ❤➭♠ ❘❛❞♦♥ ✲ ◆✐❦♦❞②♠ ❚➢➡♥❣ tù ♣❤➬♥ tr➢í❝✱ t❛ ❝ị♥❣ ❝ã ❦❤➳✐ ♥✐Ư♠ ➤➵♦ ❤➭♠ ❘❛❞♦♥ ✲ ◆✐❦♦❞②♠ ❝đ❛ ❤❛✐ ➤é ➤♦ ❦❤➠♥❣ ❝é♥❣ tÝ♥❤✳ ❈❤♦ µ ✈➭ ν ♠ét σ ✲ ➤➵✐ sè ❧➭ ❤❛✐ ❤➭♠ t❐♣ U σ ✲ ❝é♥❣ tÝ♥❤ ✭❤❛✐ ➤é ➤♦✮ ➤➢ỵ❝ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥ ❝➳❝ t❐♣ ❝♦♥ ❝đ❛ ♠ét t❐♣ f : U → [0, ∞) s❛♦ ❝❤♦ µ(A) = U✳ ◆Õ✉ ❝ã ♠ét ❤➭♠ f dν ∀A ∈ U ✱ tì f U ợ ợ ọ ❤➭♠ A ❘❛❞♦♥ ✲ ◆✐❦♦❞②♠ ❝đ❛ µ ➤è✐ ✈í✐ ν ợ ết f = dà/d ❜✐Õt r➺♥❣✱ tr➢ê♥❣ ❤ỵ♣ tr➟♥ ①➯② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ µ ❧➭ t✉②Ưt ➤è✐ ❧✐➟♥ tơ❝ ➤è✐ ν ✱ ❦ý ❤✐Ư✉ µ