Nghiên cứu một số tính chất của không gian xác suất tổng quát

▲í✐ ❝↔♠ ì♥ ▲í✐ ✤➛✉ t✐➯♥ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐✿ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✲▲➼✲❚✐♥✱ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❇ë ♠æ♥ ●✐↔✐ t➼❝❤✱ ✤➦❝ t t ụ t ũ ữớ ữợ ự ữợ ụ ữ tổ õ t❤➯♠ ♥❣❤à ❧ü❝ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ t♦➔♥ t❤➸ ợ P õ ỳ ỵ õ ❣â♣✱ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❙ì♥ ▲❛✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ ❇✉♥ ❚❤➠♥ ❈➙✉ ❱➠♥❣ ▲✐❛ ❉♦ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✶ ▼ð ✤➛✉ ✹ ✶ ❍➔♠ sè ❇♦r❡❧ ✼ ✶✳✶ σ ✲ ✤↕✐ sè ❇♦r❡❧ tr♦♥❣ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❍➔♠ sè ❇♦r❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷ ❑❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✶✺ ✷✳✶ ✣ë ✤♦ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ✣à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✈➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸ ❚➼❝❤ ♣❤➙♥ ✷✷ ✸✳✶ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤ì♥ ❣✐↔♥ ❦❤æ♥❣ ➙♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷đ❝ ❦❤ỉ♥❣ ➙♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷đ❝ ❣✐→ trà ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹ ❑ý ✈å♥❣ ✈➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✸✶ ✹✳✶ ❑ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✷ ❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✸ ❙ü ✤ë❝ ❧➟♣ ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✹ ❙ü ❤ë✐ tö ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✹✳✶ ❍ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✹✳✷ ❍ë✐ tö t❤❡♦ ①→❝ s✉➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷ ❑➳t ❧✉➟♥ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✶ ✸ é ỵ õ t ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥❣➔♥❤ q✉❛♥ trå♥❣ ♥❤➜t ❝õ❛ ❚♦→♥ ❤å❝ ✈➔ ♠❛♥❣ ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ t❤ü❝ t➳ ❝✉ë❝ sè♥❣✳ ❚r♦♥❣ ❤♦↕t ✤ë♥❣ t❤ü❝ t✐➵♥✱ ❝♦♥ ♥❣÷í✐ ❜➢t ❜✉ë❝ t ú ợ tữủ ổ t ỹ trữợ ữủ ữớ ❝â t❤➸ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤➺ t❤è♥❣ ❤â❛ ❝→❝ ❤✐➺♥ t÷đ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤➸ rót r❛ ❝→❝ q✉② ❧✉➟t ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❜✐➸✉ ❞✐➵♥ ❝❤ó♥❣ ❜➡♥❣ ❝→❝ ♠ỉ ❤➻♥❤ ❚♦→♥ ❤å❝✳ ❚ø ✤â ♠ët ❧➽♥❤ ✈ü❝ ❝õ❛ ❚♦→♥ ❤å❝ ♠❛♥❣ t ỵ tt st r ♥❣❤✐➯♥ ❝ù✉ ❝→❝ q✉② ❧✉➟t ✈➔ q✉② t➢❝ t➼♥❤ t♦→♥ tữủ ỵ tt st r ✤í✐ ✈➔♦ ♥û❛ ❝✉è✐ t❤➳ ❦✛ ❳❱■■✳ ▼ët sè ♥❤➔ ❚♦→♥ ❤å❝ ♥❤÷ ❍✉②❣❡♥s✱ ❇❡r♥♦✉❧❧✐✱ ❉❡ ▼♦✐✈r❡ ❧➔ ♥❤ú♥❣ ♥❣÷í✐ ❝â ❝ỉ♥❣ ✤➛✉ t✐➯♥ t↕♦ ♥➯♥ ❝ì sð ❚♦→♥ ❤å❝ ỵ tt st s ✶✽✾✹✮✱ ❇♦r❡❧✭✶✽✼✶ ✲ ✶✾✺✻✮✱ ❑♦❧♠♦❣♦r♦✈✭✶✾✵✸ ✲ ✶✾✽✼✮✱✳✳✳ ✤➣ ❝â ♥❤✐➲✉ õ õ t ợ sỹ t tr ỵ tt st ỵ tt st trð t❤➔♥❤ ♠ët ♥❣➔♥❤ ❚♦→♥ ❤å❝ ❧ỵ♥✱ ❝❤✐➳♠ ✈à tr➼ q trồ ỵ tt ự õ ✤÷đ❝ ù♥❣ ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❑❤♦❛ ❤å❝ ❦➽ t❤✉➟t✱ ❑❤♦❛ ❤å❝ ①➣ ❤ë✐ ✈➔ ◆❤➙♥ ✈➠♥✳ ❚ø ✤â ●✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐ trð t❤➔♥❤ ♠ët ❝ỉ♥❣ ❝ư ❤ú✉ ➼❝❤ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ ❝õ❛ ỵ tt st t s ỡ ởt sè ✈➜♥ ✤➲ ✈➲ ①→❝ s✉➜t tr♦♥❣ ●✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐✱ ❡♠ ✤➣ ❝❤å♥ ✤➲ t➔✐✿ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t ✤➸ ❧➔♠ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤ ♥❤➡♠ t➻♠ ❤✐➸✉ ❤✐➺✉ q✉↔ ❤ì♥ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t✳ ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✲ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➔ →♣ ❞ư♥❣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t ✈➔ ❧➔♠ s→♥❣ tä ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✲ ❘➧♥ ❧✉②➺♥ ❦❤↔ ♥➠♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝õ❛ ❜↔♥ t ố tữủ ự ự ỵ tt ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t✳ ✹ ✹✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ❚➻♠ ❤✐➸✉✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✲ ❙÷✉ t➛♠✱ ✤å❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤ tê♥❣ ❤ñ♣ ❝→❝ ❦✐➳♥ t❤ù❝✳ ✲ r t ợ ữợ tr ụ ữ sr ợ tờ ổ ữợ õ õ ứ õ tờ ❤đ♣ ❦✐➳♥ t❤ù❝ ✈➔ tr➻♥❤ ❜➔② t❤❡♦ ✤➲ ❝÷ì♥❣ ♥❣❤✐➯♥ ❝ù✉✱ q✉❛ ✤â t❤ü❝ ❤✐➺♥ ❦➳ ❤♦↕❝❤ ✈➔ ❤♦➔♥ t❤➔♥❤ õ ợ ữợ t tr ❦❤â❛ ❧✉➟♥ ✻✳✶✳ ❚➼♥❤ ♠ỵ✐ ♠➫ ❝õ❛ ❦❤â❛ ❧✉➟♥ ✣➙② ❧➔ ♠ët ✈➜♥ ✤➲ ❦❤→ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❜↔♥ t❤➙♥ tr t ỗ tớ ụ ♠ët ✈➜♥ ✤➲ ❝á♥ ❝❤÷❛ ✤÷đ❝ t✐➳♣ ❝➟♥ ♥❤✐➲✉ ✤è✐ ✈ỵ✐ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✣❍❙P ❚♦→♥ ❤✐➺♥ ♥❛② t↕✐ trữớ ữợ t tr õ tư❝ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t✳ ✼✳ ◆❤ú♥❣ ✤â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ✤➣ tê♥❣ ❤đ♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ì ❜↔♥ ✤➛② ✤õ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t tê♥❣ q✉→t✳ ✽✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ♥❤÷ ✈➟② ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ữỡ ợ ỳ s ữỡ ✶✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❜❛♥ ✤➛✉ ✈➲ ❣✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐ ♥❤÷ ❤➔♠ sè ❇♦r❡❧ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ õ ữỡ ữợ ự ổ ❣✐❛♥ ①→❝ s✉➜t✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❈❤÷ì♥❣ ✸✿ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ❝õ❛ ❤➔♠ ✤♦ ✤÷đ❝ ❝ơ♥❣ ♥❤÷ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷đ❝ ❣✐→ trà ♣❤ù❝ ❝➛♥ t❤✐➳t ❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳ ✣➦❝ ❜✐➺t✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② t tr ỡ ỳ ỵ tt t ✺ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ❤ú✉ ❤↕♥ (X, F, µ) ❜➜t ❦➻✳ ❈❤÷ì♥❣ ✹✿ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➲ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ♠ët ✤➦❝ tr÷♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧➔ ❦ý ✈å♥❣✳ ◆❣♦➔✐ r❛ tr♦♥❣ ♣❤➛♥ ❝✉è✐ ❝❤÷ì♥❣✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➲ sü ❤ë✐ tö ❝õ❛ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ✻ ❈❤÷ì♥❣ ✶ ❍➔♠ sè ❇♦r❡❧ ❈❤÷ì♥❣ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ✈➲ σ ✲ ✤↕✐ sè ❇♦r❡❧ tr♦♥❣ R ✈➔ ❤➔♠ sè ❇♦r❡❧✳ ✶✳✶ σ✲ ✤↕✐ sè ❇♦r❡❧ tr♦♥❣ R ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ●✐↔ sû C ❧➔ t➟♣ ❤ñ♣ ❝→❝ t➟♣ ❝♦♥ ♠ð tr♦♥❣ R✳ ❑❤✐ ✤â F(C) ✤÷đ❝ ❣å✐ ❧➔ σ✲ ✤↕✐ sè ❇♦r❡❧ tr♦♥❣ R✱ t❤÷í♥❣ ✤÷đ❝ ✈✐➳t t➢t ❧➔ B(R)✳ ❈→❝ t➟♣ ♥➡♠ tr♦♥❣ B(R) ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ t➟♣ ❇♦r❡❧✳ ◆❤÷ ✈➟② B(R) ❧➔ σ✲ ✤↕✐ sè s✐♥❤ ❜ð✐ ❝→❝ t➟♣ ❝♦♥ ♠ð tr♦♥❣ R✳ ▼➺♥❤ ✤➲ ✶✳✶✳✷✳ ❈→❝ t➟♣ ❝♦♥ s❛✉ ✤➙② tr♦♥❣ R t❤✉ë❝ B(R)✿ ✐✮ C1 = (a, b) ✈ỵ✐ ❜➜t ❦➻ a < b❀ ✐✐✮ C2 = (−∞, a) ✈ỵ✐ ❜➜t ❦➻ a ∈ R❀ ✐✐✐✮ C3 = (a, ∞) ✈ỵ✐ ❜➜t ❦➻ a ∈ R❀ ✐✈✮ C4 = [a, b] ✈ỵ✐ ❜➜t ❦➻ a ≤ b❀ ✈✮ C5 = (−∞, a] ✈ỵ✐ ❜➜t ❦➻ a ∈ R❀ ✈✐✮ C6 = [a, ∞) ✈ỵ✐ ❜➜t ❦➻ a ∈ R❀ ✈✐✐✮ C7 = (a, b] ✈ỵ✐ ❜➜t ❦➻ a < b❀ ✈✐✐✐✮ C8 = [a, b) ✈ỵ✐ ❜➜t ❦➻ a < b❀ ✐①✮ ❚➟♣ ❝♦♥ ✤â♥❣ ❜➜t ❦➻ tr♦♥❣ R ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ♥❤➟♥ t❤➜② ❝→❝ t➟♣ C1, C2, C3 ❧➔ ♥❤ú♥❣ t➟♣ ♠ð ✈➔ ❞♦ ✤â t❤✉ë❝ B(R)✳ ❚❛ ❝â ∞ C4 = [a, b] = (a − n1 , b + n1 ) ∈ B(R), n=1 ∞ C5 = (−∞, a] = n=1 (−∞, a + n1 ) ∈ B(R), ✼ ∞ C6 = [a, ∞) = (a − n1 , ∞) ∈ B(R), n=1 ∞ C7 = (a, b] = (a, b + n1 ) ∈ B(R), n=1 ∞ C8 = [a, b) = (a − n1 , b) ∈ B(R) n=1 ●✐↔ sû K ❧➔ ♠ët t➟♣ ✤â♥❣ tr♦♥❣ R✱ ❦❤✐ ✤â K c ∈ B(R) ▼➔ t❛ ❧↕✐ ❝â K = (K c )c ∈ B(R)✳ ❉♦ ✤â t➟♣ ❝♦♥ ✤â♥❣ ❜➜t ❦➻ tr♦♥❣ R ❝ô♥❣ t❤✉ë❝ B(R) ▼➺♥❤ ✤➲ ✶✳✶✳✸✳ ❈❤♦ F ✭✤â♥❣✮ ❧➔ σ✲ ✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ tr♦♥❣ R s✐♥❤ ❜ð✐ ❝→❝ t➟♣ ❝♦♥ ✤â♥❣ tr♦♥❣ R ✈➔ F ✭❝♦♠♣❛❝t✮ ❧➔ σ✲ ✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ tr♦♥❣ R s✐♥❤ ❜ð✐ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t tr♦♥❣ R ❑❤✐ ✤â t❛ ❝â F ✭✤â♥❣✮❂F ✭❝♦♠♣❛❝t✮❂B(R)✳ ❈❤ù♥❣ ♠✐♥❤✳ ▼å✐ t➟♣ ❝♦♥ ✤â♥❣ tr♦♥❣ R ✤➲✉ t❤✉ë❝ B(R)✳ ❉♦ F ✭✤â♥❣✮ ❧➔ σ✲ ✤↕✐ sè ♥❤ä ♥❤➜t ❝❤ù❛ ❝→❝ t➟♣ ♥➯♥ t❛ ♣❤↔✐ ❝â F ✭✤â♥❣✮⊆ B(R)✳ ▼➦t ❦❤→❝ ♠é✐ t➟♣ ♠ð ❧➔ ♣❤➛♥ ❜ò ❝õ❛ ♠ët t➟♣ ✤â♥❣ ✈➔ ❞♦ ✤â t❤✉ë❝ F ✭✤â♥❣✮✳ ▼➔ t❛ ❧↕✐ ❝â B(R) ❧➔ σ ✲ ✤↕✐ sè s✐♥❤ ❜ð✐ ❝→❝ t➟♣ ♠ð tr♦♥❣ R ♥➯♥ t❛ ❝â B(R) ⊆ F ✭✤â♥❣✮✳ ❚ø ✤â t❛ ❝â F ✭✤â♥❣✮❂B(R) ❇➙② ❣í t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ F ✭✤â♥❣✮❂F ✭❝♦♠♣❛❝t✮✳ ❉♦ ♠é✐ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ R ✤➲✉ ❧➔ t➙♣ ✤â♥❣ ♥➯♥ t❛ ❝â F ✭❝♦♠♣❛❝t✮⊆ F ✭✤â♥❣✮✳❚✉② ♥❤✐➯♥ ❜➜t ❝ù t➟♣ ✤â♥❣ K ♥➔♦ ❝ơ♥❣ ❝â t❤➸ ✈✐➳t ✤÷đ❝ ❞↕♥❣ ❤đ♣ ✤➳♠ ✤÷ì❝ ♥❤÷ s❛✉✿ ∞ [−n, n] ∩ K K= n=1 ❚❛ t❤➜② ♠é✐ t➟♣ [−n, n]∩K ✤â♥❣ ✈➔ ❜à ❝❤➦♥ ❞♦ ✤â ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❚ø ✤â K ∈ F(compact) ✈➔ ❞♦ ✤â F ✭✤â♥❣✮⊆ F ✭❝♦♠♣❛❝t✮✳ ◆❤÷ ✈➟② t❛ ❝â✿ F ✭✤â♥❣✮ ⊆ F ✭❝♦♠♣❛❝t✮ = B(R) ✶✳✷ ❍➔♠ sè ❇♦r❡❧ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ (X, F) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤♦✱ f : X → R ❧➔ ♠ët ❤➔♠ sè✳ ❑❤✐ ✤â f ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ f −1(G) ∈ F ✈ỵ✐ G ❧➔ t➟♣ ♠ð tr♦♥❣ R✳ ✽ ▼➺♥❤ ✤➲ ✶✳✷✳✷✳ ❍➔♠ f : X → R ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f −1(A) ∈ F ✈ỵ✐ ♠é✐ A ∈ B(R) ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f −1(A) ∈ F ✈ỵ✐ ♠é✐ A ∈ B(R) ❑❤✐ ✤â t❛ ❝â f −1 (G) ∈ F ✈ỵ✐ ♠é✐ t➟♣ ♠ð G tr♦♥❣ R✱ ❞♦ ✤â f ❧➔ ❤➔♠ ❇♦r❡❧✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû f −1 (G) ∈ F ✈ỵ✐ ♠é✐ t➟♣ ♠ð G tr♦♥❣ R✳ ❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ f −1 (A) ∈ F ✈ỵ✐ ♠é✐ A ∈ B(R) ❚❤➟t ✈➟②✱ t❛ ✤➦t S = {E ⊆ R : f −1 (E) ∈ F} ❧➔ t➟♣ ❤ñ♣ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ R✳ S ❧➔ ♠ët σ ✲ ✤↕✐ sè✱ t❤➟t ✈➟② t❛ ❝â✿ ✐✮ f −1 (R) = X ∈ F ❞♦ ✤â R ∈ S ✳ ✐✐✮ f −1 (E) ∈ F ♥➯♥ X \ f −1 (E) = f −1 (R \ E) ∈ F ✱ ❞♦ ✤â R \ E ∈ S ✳ ✐✐✐✮ ●✐↔ sû E1 , E2 , ∈ S, ❦❤✐ ✤â f −1 (E1 ), f −1 (E2 ), ∈ F ✱ s✉② r❛ f −1 (E1 ) ∪ f −1 (E2 ) ∪ = f −1 (E1 ∪ E2 ∪ ) ∈ F ❉♦ ✤â t❛ ❝â E1 ∪ E2 ∪ ∈ S ✳ S ❧➔ σ ✲ ✤↕✐ sè ❝❤ù❛ ❝→❝ t➟♣ ♠ð✱ tø ✤â B(R) ⊆ S ✳ ❉♦ ✤â t❛ ❝â f −1 (A) ∈ F ✈ỵ✐ A ∈ B(R)✳ ▼➺♥❤ ✤➲ ✶✳✷✳✸✳ ❈❤♦ C ❧➔ t➟♣ ❤ñ♣ ❝→❝ t➟♣ ❝♦♥ tr♦♥❣ R t❤ä❛ ♠➣♥ F(C) = B(R) ✈➔ ❤➔♠ f : X → R✳ ❑❤✐ ✤â f ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f −1(A) ∈ F ✈ỵ✐ ♠å✐ A ∈ C ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f ❧➔ ❤➔♠ ❇♦r❡❧✳ ❑❤✐ C ⊆ B(R) t❛ ❝â f −1(A) ∈ F ✈ỵ✐ ❜➜t ❦➻ A ∈ C ✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû f −1 (A) ∈ F ✈ỵ✐ ♠å✐ A ∈ C ✳ ✣➦t S = {E ⊆ R : f −1 (E) ∈ F} ❧➔ σ ✲ ✤↕✐ sè ❝❤ù❛ C ✳ ❚❛ ❝â B(R) = F(C) ⊆ S ✱ tù❝ ❧➔ t❛ ❝â f −1 (A) ∈ F, ∀A ∈ B(R) ◆❤➟♥ ①➨t ✶✳✷✳✹✳ ❚❛ ❝â t❤➸ ❝❤å♥ C ❧➔ ♠ët t➟♣ ❜➜t ❦➻ tr♦♥❣ ▼➺♥❤ ✤➲ ✶✳✶✳✷✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ t❛ ❝â t❤➸ ♥â✐ r➡♥❣ f ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f −1 ((−∞, a]) ∈ F ✈ỵ✐ ♠é✐ a ∈ R✳ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳ ❈❤♦ f : X → R ❧➔ ❤➔♠ ❇♦r❡❧ ✈➔ g : R → R ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝✳ ❑❤✐ ✤â g◦f :X →R ❧➔ ❤➔♠ ❇♦r❡❧✳ ✾ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ G ❧➔ ♠ët t➟♣ ♠ð ❜➜t ❦➻ tr♦♥❣ R✳ ❑❤✐ ✤â g−1(G) ❧➔ t➟♣ ♠ð tr♦♥❣ R ✈➔ ❞♦ ✤â g −1 (G) ∈ B(R)✳ ▲↕✐ ❞♦ f : X → R ❧➔ ❤➔♠ ❇♦r❡❧ ♥➯♥ f −1 (g −1 (G)) ∈ F ✈ỵ✐ g −1 (G) ∈ B(R)✳ ▼➔ t❛ ❧↕✐ ❝â (g ◦ f )−1 (G) = f −1 (g −1 (G)) ♥➯♥ (g ◦ f )−1 (G) ∈ F ✳ ❉♦ ✤â g ◦ f ❧➔ ❤➔♠ ❇♦r❡❧✳ ▼➺♥❤ ✤➲ ✶✳✷✳✻✳ ●✐↔ sû f : X → R ✈➔ g : X → R ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧✳ ❦❤✐ ✤â t➟♣ E = {x ∈ X : f (x) < g(x)} ữủ ự ợ ộ r ∈ Q✱ t❛ ✤➦t Er = {x ∈ X : f (x) < r < g(x)}✳ ❑❤✐ ✤â Er = {x : f (x) < r} ∩ {x : r < g(x)} ❧➔ ❣✐❛♦ ❝õ❛ ❤❛✐ t➟♣ ✤♦ ✤÷đ❝ tr♦♥❣ X ✈➔ ✤♦ ✤â Er ✤♦ ✤÷đ❝✳ ▼➔ E = Er ❧➔ ❤đ♣ ✤➳♠ r∈Q ✤÷đ❝ ❝õ❛ ❝→❝ t➟♣ ✤♦ ✤÷đ❝ ♥➯♥ E ✤♦ ✤÷đ❝✳ ▼➺♥❤ ✤➲ ✶✳✷✳✼✳ ❈❤♦ (X, F) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤♦ ✈➔ f : X → R, g : X → R ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧✳ ❑❤✐ ✤â ✐✮ af + b ❧➔ ❤➔♠ ❇♦r❡❧ ✈ỵ✐ ❜➜t ❦➻ a, b ∈ R❀ ✐✐✮ f + g ❧➔ ❤➔♠ ❇♦r❡❧❀ ✐✐✐✮ |f |α ❧➔ ❤➔♠ ❇♦r❡❧ ✈ỵ✐ ❜➜t ❦➻ α ≥ 0❀ ✐✈✮ ◆➳✉ f ❦❤ỉ♥❣ ❜à tr✐➺t t✐➯✉ t❤➻ f1 ❧➔ ❤➔♠ ❇♦r❡❧❀ ✈✮ f g ❧➔ ❤➔♠ ❇♦r❡❧❀ ✈✐✮ |f |✱max{f, g}✱min{f, g} ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧✳ ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❱ỵ✐ ❜➜t ❦➻ c ∈ R t❛ ①➨t t➟♣ A = {x ∈ X : (af + b)(x) ≤ c} = {x : af (x) + b ≤ c} = {x : af (x) ≤ c − b} =  {x : f (x) ≤    {x : f (x) ≥  X  ∅ ✶✵ c−b a }, c−b a }, ❛❃✵ ❛❁✵ a = 0, c ≥ b a = 0, c < b ❑❤✐ ✤â t❛ ❝â +∞ E(g(X)) = g(x)e −(x−µ)2 2σ dx √ σ 2π −∞ ✣à♥❤ ♥❣❤➽❛ ✹✳✶✳✽✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ ♥➳✉ ❤➔♠ ♣❤➙♥ ♣❤è✐ FX ❝õ❛ ♥â ❝â ❞↕♥❣ x FX (x) = ϕ(t)dt, −∞ tr♦♥❣ ✤â ϕ ❧➔ ❤➔♠ ❦❤æ♥❣ ➙♠ tr➯♥ R t❤ä❛ ♠➣♥ ϕ(t)dt = ❚❛ ❝â t❤➸ ♥â✐ r➡♥❣ X ❧✐➯♥ R tö❝ t❤❛② ✈➻ X ❧✐➯♥ tö❝ t✉②➺t ✤è✐✳ ◆➳✉ X ❧✐➯♥ tö❝✱ ❦❤✐ ✤â FX : R → [0, 1] ❝✉♥❣ ❧✐➯♥ tö❝✳ ❚❤➟t ✈➟②✱ ❝è ✤à♥❤ x ∈ R✱ ❣✐↔ sû an < x ✈ỵ✐ ♠å✐ n ✈➔ an → x ❚❛ ❝â F(x) (x) − F(an ) (x) = ϕ(t)dt = (an ,x] I(an ,x] (t)ϕ(t)dt R ✣➦t gn (t) = I(an ,x] (t)ϕ(t)✱ ❦❤✐ ❞â ≤ gn (t) ≤ ϕ(t) ✈➔ gn (t) → I{x} (t)(t) ỵ s sỹ tö ❜à ❝❤➦♥✱ t❛ ❝â gn (t)dt → R I{x} (t)ϕ(x)dt = R ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä FX ❧✐➯♥ tö❝ tr→✐ t↕✐ x✱ ♠➦t ❦❤→❝ Fx ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ♥➯♥ FX ❧✐➯♥ tö❝ ♣❤↔✐✱ tø ✤â t❛ ❦➳t ❧✉➟♥ r➡♥❣ FX ❧✐➯♥ tö❝ tr➯♥ R✳ ✹✳✷ ❍➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✣à♥❤ ♥❣❤➽❛ ✹✳✷✳✶✳ ❈❤♦ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t (Ω, S, P)✳ ❑❤✐ ✤â ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ X ❧➔ ❤➔♠ ϕX : R → C ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ eitX dP = E(eitX ) = E(cos(tX)) + iE(sin(tX)), t ∈ R ϕX (t) = Ω cos(tX) ✈➔ sin(tX) ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧ ❜à ❝❤➦♥ tr➯♥ Ω ✈➔ ❞♦ ✤â t ợ t R ỵ ❍➔♠ ✤➦❝ tr÷♥❣ ϕX t❤ä❛ ♠➣♥✿ ✐✮ |ϕX (t)| ≤ ϕX (0) = 1, ✸✼ ✐✐✮ ϕX ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ R✱ ✐✐✐✮ ϕX (t) = ϕX (−t) ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❚❛ ❝â ϕX (0) = 1✱ ❤ì♥ ♥ú❛ eitX dP| ≤ |ϕX (t)| = | Ω |eitX |dP = Ω dP = Ω ✐✐✮ ❱ỵ✐ s, t ❜➜t ❦➻ t❛ ❝â (eitX − eisX )dP| |ϕX (t) − ϕX (s)| = | Ω |eitX − eisX |dP ≤ Ω |eisX ||ei(t−s)X − 1|dP = Ω |ei(t−s)X − 1|dP = ữ ợ ộ t õ |eiδn X(ω) − 1| → ❦❤✐ n → ∞, n ỵ s sỹ tö ❜à ❝❤➦♥✱ t❛ ❦➳t ❧✉➟♥ r➡♥❣ ϕX ❧✐➯♥ t✉❝ ✤➲✉ tr➯♥ R ✐✐✐✮ ❚❛ ❝â ϕX (t) = E(cos(tX)) + iE(sin(tX)) = E(cos(tX)) − iE(sin(tX)) = E(cos(−tX)) + iE(sin(−tX)) = X (t) ỵ sỷ tỗ t E(|X|n) ✈ỵ✐ n ✭♥❣❤➽❛ ❧➔ X n ∈ (r) ≤ r ≤ n✱ ✤↕♦ ❤➔♠ ❝➜♣ r ❝õ❛ ϕX (t) X (t) tỗ t (r) ≥ (iX)r eitX dP = E((iX)r eitX ) ϕX (t) = Ω ✈➔ E(X r ) = ϕ i (0) (r) X r ✸✽ L1 ), ❦❤✐ ✤â ✈ỵ✐ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû X n ∈ L1✱ ❦❤✐ ✤â ✈ỵ✐ r ≤ n✱ t❛ ❝â X r ∈ L1✳ ❚❛ ❝â eitX − ϕX (t + h) − ϕX (t) = E(eitX ( )) h h itX ✈➔ | e −1 h | ≤ X(ω) ✈ỵ✐ ♠é✐ ω ∈ Ω ❍ì♥ ♥ú❛ ♥➳✉ hn → ❦❤✐ n → ∞ t❤➻ t❛ ❝â eihn X(ω) − → X(ω) hn ❑❤✐ n → ∞✳ ❚❤❡♦ ✣à♥❤ ỵ s sỹ tử t õ eitX ( eihn X−1 )dP → hn Ω ❤❛② E(eitX ( e ihn X−1 hn iXeitX dP Ω )) → E(iXeitX ) ❦❤✐ n → ∞✳ (1) ❉♦ ✤â ϕX (t) ❦❤↔ ✈✐ ✈➔ t❛ ❝â ϕX (t) = E((iX)eitX )✳ ✣➦t t = 0✱ ❦❤✐ ✤â t❛ ❝â (r) ϕ (0) E(X ) = X r i r ◆❤➟♥ ①➨t ✹✳✷✳✹✳ ❚❛ ❝â t❤➸ ✈✐➳t ϕX (t) = E(eitX ) = eitX dP = Ω eitX dFX , R ❦❤✐ ✤â (n) n n E(X ) = x dFX ϕ (0) = Xn i R ❱➼ ❞ö ✹✳✷✳✺✳ ✶✳ ◆➳✉ X ❝â ♣❤➙♥ ♣❤è✐ ♥❤à t❤ù❝ ✈ỵ✐ t❤❛♠ sè (n, p) t❤➻ n n itk ϕX (t) = e Cnk pk q n−k Cnk (peit )k q n−k = (peit + q)n = k=0 k=0 ✷✳ ◆➳✉ X ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ t❤❡♦ t❤❛♠ sè µ t❤➻ ∞ ϕX (t) = e k=0 itk µ k e−µ k! ∞ =e −µ k=0 it it−1 (µeit )k = e−µ eµe = eµ(e ) k! ✸✳ ◆➳✉ X ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ N (µ, σ2) t❤➻ ϕX (t) = eitµe −σ t2 ❚✐➳♣ t❤❡♦ ✤➙②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ sü ✤ë❝ ❧➟♣ ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ✈ỵ✐ ❤❛✐ ❞↕♥❣ ❤ë✐ tư ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✿ ❤ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥✱ ❤ë✐ tö t❤❡♦ ①→❝ s✉➜t✳ ✣➙② ❧➔ ❤❛✐ ❞↕♥❣ ❤ë✐ tö ✤â♥❣ ✈❛✐ trá t❤❡♥ ❝❤èt tr♦♥❣ ▲✉➟t sè ❧ỵ♥✳ ✸✾ ✹✳✸ ❙ü ✤ë❝ ❧➟♣ ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✣à♥❤ ♥❣❤➽❛ ✹✳✸✳✶✳ ❈→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X1, , Xn ✤÷đ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣ ♥➳✉ ✈ỵ✐ ❝→❝ t➟♣ ❇♦r❡❧ E1, En ❜➜t ❦➻ tr♦♥❣ R t❛ ❝â P(X1 ∈ E1 , , Xn ∈ En ) = P(X1 ∈ E1 ) P(Xn ∈ En ) tù❝ ❧➔ n P(X1−1 (E1 ) ∩ ∩ Xn−1 (En )) P(Xj−1 (Ej )) = j=1 ❉♦ ✤â ❤❛✐ ❜✐➳♥ ❝è A ✈➔ B ✤ë❝ ❧➟♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ IA IB ỵ ♥❣➝✉ ♥❤✐➯♥ X1, , Xn ✤ë❝ ❧➟♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ P(X1 ≤ a1 , , Xn ≤ an ) = P(X1 ≤ a1 ) P(Xn ≤ an ) tr♦♥❣ ✤â a1, , an ∈ R✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû X1, , Xn ✤ë❝ ❧➟♣✱ ❦❤✐ ✤â t❤❡♦ ✣à♥❤ ♥❣❤➽❛ 4.3.1 t❛ ❝â P(X1 ∈ E1 , , Xn ∈ En ) = P(X1 ∈ E1 ) P(Xn ∈ En ) ❚❛ ✤➦t E1 = (−∞, a1 ] , , En = (−∞, an ]✱ ❦❤✐ ✤â t❛ ❝â P(X1 ≤ a1 , , Xn ≤ an ) = P(X1 ≤ a1 ) P(Xn ≤ an ) ◆❣÷đ❝ ❧↕✐✱ ●✐↔ sû t❛ ❝â (4.1)✱ ❝è ✤à♥❤ a1 , , an ✳ ❱ỵ✐ ❜➜t ❦➻ t➟♣ E ∈ B(R) t❛ ✤➦t µ(E) = P(X1 ∈ E, X2 ≤ a2 , , Xn ≤ an ), ν(E) = P(X1 ∈ E)P(X2 ≤ a2 ) P(Xn an ) õ ợ ✈➔ ν ❧➔ ❝→❝ ✤ë ✤♦ ❤ú✉ ❤↕♥ tr➯♥ B(R)✳ ❍ì♥ ♥ú❛ ✈ỵ✐ ❜➜t ❦➻ a ∈ R t❛ ❝â µ((−∞, a]) = P(X1 ≤ a, X2 ≤ a2 , , Xn ≤ an ) = P(X1 ≤ a)P(X2 ≤ a2 ) P(Xn ≤ an ) = ν((−∞, a]) ✹✵ ữ t õ à((, a]) = ((, a]) a t ữủ à(R) = (R) õ ợ t , t õ à((, ]) = µ((−∞, β]) − µ((−∞, α]) = ν((−∞, β]) − ν((−∞, α]) = ν((α, β]) ❍ì♥ ♥ú❛✱ µ((a, ∞)) = µ(R) − µ((−∞, a]) = ν(R) − ν((−∞, a]) = ν((a, ∞)) ●✐↔ sû A ❧➔ ✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ tr♦♥❣ R s✐♥❤ ❜ð✐ ❦❤♦↔♥❣ (−∞, a] ✈ỵ✐ a ∈ R ❑❤✐ ✤â ♥➳✉ ✤➦t I(a) = (−∞, a] , J(a, b) = (a, b] ✈ì✐ a < b t A ỗ t õ ữ s❛✉ R, ∅, I(a1 ) ∪ J(a2 , b2 ) ∪ ∪ J(an , bn ), J(a1 , b1 ) ∪ ∪ (am , bm ), J(a1 , b1 ) ∪ ∪ J(ak , bk ) ∪ (α, ∞), I(a1 ) ∪ J(a2 , b2 ) ∪ ∪ J(ar , br ) ∪ (α, ∞) ❈→❝ t➟♣ tr♦♥❣ ♠é✐ ❤đ♣ ❤ú✉ ❤↕♥ ❝â t❤➸ rí✐ ♥❤❛✉✳ õ t õ à(E) = (E) ợ E A ❜➜t ❦➻ ❇➙② ❣✐í t❛ ✤➦t M = {E ∈ B(R) : µ(E) = ν(E)} ◆➳✉ A1 ⊆ A2 ⊆ tr♦♥❣ M t❤➻ t❛ ❝â µ Ai i ❉♦ ✤â = lim µ(An ) = lim ν(An ) = ν n n Ai i Ai ∈ M i ❚÷ì♥❣ tü✱ ♥➳✉ B1 ⊇ B2 ⊇ ⊇ Bn ⊇ tr♦♥❣ M t❤➻ t❛ ❝ô♥❣ ❝â Bi ∈ M✳ ❚ø ✤â t❛ ❝â i M ❧➔ ♠ët ❧ỵ♣ ✤ì♥ ✤✐➺✉✳ ❑❤✐ A ⊆ M t❛ ❝â F(A) ⊆ M ⊆ B(R) ✹✶ ▼➔ t❛ ❝â F(A) = B(R) ♥➯♥ M = B(R)✳ ❇➙② ❣✐í ❝è ✤à♥❤ E1 tr♦♥❣ B(R) ợ ộ E B(R) t t à(E) = P(X1 ∈ E1 , X2 ∈ E, X3 ≤ a3 , , Xn ≤ n) ν(E) = P(X1 ∈ E1 )P(X2 ∈ E)P(X3 ≤ a3 ) P(Xn ≤ n) ❚❤❡♦ tr➯♥ t❛ ❝â µ = ν tr➯♥ A ✈➔ t❛ ❝â µ = ν tr➯♥ B(R) = F(A)✳ ✣à♥❤ ♥❣❤➽❛ ✹✳✸✳✸✳ ●✐↔ sû t❛ ❝â X1, , Xn ❧➔ ❝→❝ õ ố ỗ tớ ❝õ❛ X1, , Xn ❧➔ FX1 , ,Xn (x1 , , xn ) = P(X1 ≤ x1 , , Xn ≤ xn ) ◆❤➟♥ ①➨t ✹✳✸✳✹✳ FX , ,X n ❝ơ♥❣ ❝â t➼♥❤ ❝❤➜t t÷ì♥❣ tü ♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ FX ✱ ❝❤➥♥❣ ❤↕♥ t➼♥❤ ❝❤➜t ✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tö❝ ♣❤↔✐✳ ❍➺ q✉↔ ✹✳✸✳✺✳ ❈→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X1, , Xn ✤ë❝ ❧➟♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ FX1 , ,Xn (x1 , , xn ) = FX1 (x1 ) FXn (x1 ), tr♦♥❣ ✤â x1 , , xn R ỵ sỷ X1, , Xn ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✈➔ g1, , gn ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧ tr➯♥ R✳ ❑❤✐ ✤â ❝→❝ ❤➔♠ ♥❣➝✉ ♥❤✐➯♥ Y1 = g1 (X1 ), , Yn = gn (Xn ) ❝ô♥❣ ✤ë❝ ❧➟♣ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû E1, , En ❧➔ ❝→❝ t➟♣ ❇♦r❡❧ tr♦♥❣ R✳ ❑❤✐ ✤â P(Y1 ∈ E1 , , Yn ∈ En ) = P(g1 (X1 ) ∈ E1 , , gn (Xn ) ∈ En ) = P(X1 ∈ g1−1 (E1 ), , Xn ∈ gn−1 (En )) = P(X1 ∈ g1−1 (E1 )) P(Xn ∈ gn−1 (En )) = P(Y1 ∈ E1 ) P(Yn ∈ En ) ✳ ✹✷ trữ ỗ tớ n ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X1, , Xn ❧➔ ❤➔♠ ϕX1 , ,Xn : Rn → C ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ϕX1 , ,Xn (t1 , , tn ) = E(ei(t1 X1 + +tn Xn ) ) ✈ỵ✐ (t1, , tn) ∈ Rn ỵ trữ ỗ tớ t ởt ố ỗ tớ ỵ X1, , Xn ❧➟♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ E(g1 (X1 )) gn (Xn )) = E(G1 (X1 )) E(gn (Xn )) ✭✹✳✷✮ tr♦♥❣ ✤â gi : R → C, i = 1, n t❤ä❛ ♠➣♥ g1(X1), , gn(Xn) ❦❤↔ t➼❝❤ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû E1, , En ❧➔ ❝→❝ t➟♣ ❇♦r❡❧ tr♦♥❣ R✳ ❚❛ ✤➦t g1 = IE1 , ,En = IEn ▲ó❝ ♥➔②✱ gi : R → R ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧ ❜à ❝❤➦♥✱ ❞♦ ✤â gi (Xi ) ❦❤↔ t➼❝❤ ✈ỵ✐ i = 1, n✳ ❇➙② ❣✐í ❣✐↔ sû ❝â (4.2)✱ ❦❤✐ ✤â E(IE1 (X1 ) IEn (Xn )) = E(IE1 (X1 )) E(IEn (Xn )), tr♦♥❣ ✤â IE1 (X1 ) IEn (Xn ) : Ω → R ❧➔ ❤➔♠ ♥❤➟♥ ❣✐→ trà ❜➡♥❣ ♥➳✉Xi ∈ Ei ✈➔ ❜➡♥❣ ♥➳✉ Xi ∈ / Ei , i = 1, n✳ ❉♦ ✤â ❦ý ✈å♥❣ ❝õ❛ ❤➔♠ ♥➔② ❧➔ E(IE1 (X1 ) IEn (Xn )) = 1.P(X1 ∈ E1 , , Xn ∈ En ) ❚÷ì♥❣ tü✱ E(IEi (Xi )) = 1.P(Xi ∈ Ei ) ✈ỵ✐ i = 1, n✳ ❉♦ ✤â (4.2) trð t❤➔♥❤ P(X1 ∈ E1 , , Xn ∈ En ) = P(X1 ∈ E1 ) P(Xn ∈ En ) ♥❣❤➽❛ ❧➔ X1 , , Xn ✤ë❝ ❧➟♣✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû X1 , , Xn ✤ë❝ ❧➟♣✱ t❤❡♦ tr➯♥ t❛ ❝â E(IE1 (X1 ) IEn (Xn )) = 1.P(X1 ∈ E1 Xn ∈ En ) = P(X1 ∈ E1 ) P(Xn ∈ En ) = E(IE1 (X1 )) E(IEn (Xn )) ✹✸ ◆❤÷ ✈➟② t❛ ❝â (4.2) ✤ó♥❣ ✈ỵ✐ gi ❧➔ ❝→❝ ❤➔♠ ❝❤➾ t✐➯✉✳ ✣è✐ ✈ỵ✐ gi ❧➔ ❝→❝ ❤➔♠ ❜➜t ❦➻ s❛♦ ❝❤♦ gi (Xi ) t ỵ s sü ❤ë✐ tư ✤ì♥ ✤✐➺✉✱ t❛ ❝ơ♥❣ ❝â (4.2) ❍➺ q✉↔ ✹✳✸✳✶✵✳ ❈→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X1, , Xn ✤ë❝ ❧➟♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ϕX1 , ,Xn (t1 , , tn ) = ϕX1 (t1 ) ϕXn (tn ) ✭✹✳✸✮ ✈ỵ✐ (t1, , tn) ∈ Rn ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû X1, , Xn ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥➳✉ ✤➦t gi(Xi) = eit X i i ✈ỵ✐ i = 1, n t❤➻ t❛ ❝â ϕX1 , ,Xn (t1 , , tn ) = E(ei(t1 X1 + +tn Xn ) ) = E(eit1 X1 eitn Xn ) = E(g1 (X1 ) gn (Xn )) = E(g1 (X1 )) E(gn (Xn )) = E(eit1 X1 ) E(eitn Xn ) = ϕX1 (t1 ) ϕXn (tn ) ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sỷ trữ ỗ tớ tọ (4.3) ❣å✐ Y1 , , Yn ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ t❤ä❛ ♠➣♥ FXi = FYi , ∀i = 1, n, ❦❤✐ ✤â ϕY1 , ,Yn (t1 , , tn ) = ϕY1 (t1 ) ϕYn (tn ) = ϕX1 (t1 ) ϕXn (tn ) = ϕX1 , ,Xn (t1 , , tn ) trữ ỗ tớ t ởt ố ỗ t❤í✐ ♥➯♥ t❛ ♣❤↔✐ ❝â FY1 , ,Yn = FX1 , ,Xn ✳ ▼➦t ❦❤→❝ FY1 , ,Yn = FY1 FYn = FX1 , ,Xn ♥➯♥ t❛ ❝â FX1 , ,Xn = FX1 FXn , ♥❣❤➽❛ ❧➔ X1 , , Xn ✤ë❝ ❧➟♣✳ ✹✹ ✹✳✹ ❙ü ❤ë✐ tö ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✹✳✹✳✶ ❍ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✣à♥❤ ♥❣❤➽❛ ✹✳✹✳✶✳ ❍❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✈➔ Y ✤÷đ❝ ❣å✐ ❧➔ ❜➡♥❣ ♥❤❛✉ ❤➛✉ ❝❤➢❝ ❝❤➢♥ (h.c.c) ♥➳✉ P({ω ∈ Ω : X(ω) = Y (ω)}) = ❚❛ t❤➜② ♥➳✉ ♥❤÷ A ✈➔ B ❧➔ ❤❛✐ ❜✐➳♥ ❝è ✈ỵ✐ P(A) = P(B) = t❤➻ P(Ac ) = P(B c ) = 0✳ ❉♦ ✤â t❛ ❝â ≤ P((A ∩ B)c ) = P(Ac ∪ B c ) ≤ P(Ac ) + P(B c ) = s✉② r❛ P(A ∩ B) = 1✳ ▼➺♥❤ ✤➲ ✹✳✹✳✷✳ ●✐↔ sû ∼ ❧➔ q✉❛♥ ❤➺ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ X ∼ Y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ X = Y ❦❤✐ ✤â ∼ ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ ❤đ♣ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳ h.c.c ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ t❤➜② X ∼ X ♥➳✉ X ∼ Y t❤➻ t❛ ❝ô♥❣ ❝â Y ∼ X ✳ ❇➙② ❣✐í ❣✐↔ sû X ∼ Y ✈➔ Y ∼ V ✱ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ X ∼ V ✳ ❚❤➟t ✈➟②✱ t❛ ✤➦t A = {ω : X(ω) = Y (ω)}, B = {ω : Y (ω) = V (ω)} ❑❤✐ P(A) = P(B) = 1✳ ▼➦t ❦❤→❝ t❛ ❝â A ∩ B ⊆ {ω : X(ω) = V (ω)} ❝❤♦ ♥➯♥ t❛ ❝â P({ω : X(ω) = V (ω)}) = ♥❣❤➽❛ ❧➔ X = V h.c.c✳ ▼➺♥❤ ✤➲ ✹✳✹✳✸✳ ●✐↔ sû X ≥ ✈➔ EX = 0✱ ❦❤✐ X = h.c.c ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ n ∈ N✱ ✤➦t An = {ω : X(ω) > n1 }✳ ❑❤✐ ✤â An ∈ S, An ⊆ An+1 ✈➔ XdP ≥ = E(X) = Ω XdP ≥ n dP = P(An ) n An An ❉♦ ✤â P(An ) = ✈ỵ✐ ♠å✐ n ∈ N ▼➦t ❦❤→❝ t❛ ❝â ∞ An = {ω : X(ω) > 0} ❝❤♦ ♥➯♥ n=1 ∞ An P = P ({ω : X(ω) > 0}) = lim P(An ) = n n=1 ❉♦ ✤â t❛ ❝â P ({ω : X(ω) = 0}) = 1✱ tù❝ ❧➔ X = h.c.c ✹✺ ❍➺ q✉↔ ✹✳✹✳✹✳ ◆➳✉ ❱❛r X = t❤➻ X ❧➔ ♠ët ❤➛✉ ❝❤➢❝ ❝❤➢♥ ❤➡♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â =❱❛r X = E|X −EX|2✳ ❱ỵ✐ |X −EX|2 ≥ 0✱ t❛ s✉② r❛ |X −EX|2 = h.c.c✱ tø ✤â t❛ ❝â X − EX h.c.c✳ ✣à♥❤ ♥❣❤➽❛ ✹✳✹✳✺✳ ❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ (fn) ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ ❤ư ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✤➳♥ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ g ♥➳✉ P({ω : fn (ω) → g(ω) ❦❤✐ ♥ → ∞}) = ▼➺♥❤ ✤➲ ✹✳✹✳✻✳ ◆➳✉ fn −−−→ f ✈➔ fn −−−→ g ❦❤✐ ♥ → ∞ t❤➻ f = g h.c.c ❤✳❝✳❝ ❤✳❝✳❝ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = {ω : fn (ω) → f (ω)}, B = {ω : fn (ω) → g(ω)}, C = {ω : f (ω) → g(ω)} ❑❤✐ ✤â P(A) = P(B) = ❞♦ ✤â P(A ∩ B) = 1✳ ◆❤÷♥❣ A ∩ B ⊆ C ♥➯♥ t❛ ❝â P(C) = ❤❛② P({ω : f (ω) = g(ω)} = tù❝ ❧➔ f = g h.c.c ▼➺♥❤ ✤➲ ✹✳✹✳✼✳ ●✐↔ sû fn −−−→ f ✈➔ gn −−−→ g ❦❤✐ ♥ → ∞✳ ❑❤✐ ✤â t❛ ❝â ❤✳❝✳❝ ❤✳❝✳❝ fn + gn −−−→ f + g ❤✳❝✳❝ ✈➔ ❢ngn −−−→ f g ❦❤✐ ♥ → ∞ ❤✳❝✳❝ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = {ω : fn (ω) → f (ω)}, B = {ω : gn (ω) → g(ω)} ❑❤✐ ✤â P(A) + P(B) = ❞♦ ✤â P(A ∩ B) = ❳➨t ❤❛✐ t➟♣ C = {ω : fn (ω) + gn (ω) → f (ω) + g(ω)}, D = {ω : fn (ω)gn (ω) → f (ω)g(ω)} ✹✻ ❚❛ ♥❤➟♥ t❤➜② A ∩ B ⊆ C ✈➔ A ∩ B ⊆ D✳ ❉♦ ✤â P(C) = P(D) = 1✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä ❤✳❝✳❝ ❤✳❝✳❝ fn + gn −−−→ f + g ✈➔ ❢n gn −−−→ f g ❦❤✐ ♥ → ∞ ✹✳✹✳✷ ❍ë✐ tö t❤❡♦ ①→❝ s✉➜t ✣à♥❤ ♥❣❤➽❛ ✹✳✹✳✽✳ ❚❛ ♥â✐ r➡♥❣ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ (fn) ❤ỉ✐ tư t❤❡♦ ①→❝ s✉➜t ✤➳♥ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ f ♥➳✉ ✈ỵ✐ ♠é✐ ε > t❛ ❝â P({ω : |fn (ω) − f (ω)| ≥ ε}) → ữ ỵ r fn → f t❤❡♦ ①→❝ s✉➜t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (fn − f ) → t❤❡♦ ①→❝ s✉➜t✳ ▼➺♥❤ ✤➲ ✹✳✹✳✾✳ ◆➳✉ fn → f ✈➔ gn → g t❤❡♦ ①→❝ s✉➜t t❤➻ fn + gn → f + g t❤❡♦ ①→❝ s✉➜t ✈➔ fngn → f g t❤❡♦ ①→❝ st ự ợ > trữợ t ❝â |fn (ω) + gn (ω) − f (ω) − g(ω)| ≤ |fn (ω) − f (ω)| + |gn (ω) − g(ω)| ✈➔ ❞♦ ✤â 1 {ω : |fn (ω) − f (ω)| < ε} ∩ {ω : |gn (ω) − g(ω)| < ε} 2 ⊆ {ω : |fn (ω) + gn (ω) − f (ω) − g(ω)| < ε} ❚ø ✤â t❛ ❝â {ω : |fn (ω) + gn (ω) − f (ω) − g(ω)| ≥ ε} 1 ⊆ {ω : |fn (ω) − f (ω)| ≥ ε} ∪ {ω : |gn (ω) − g(ω)| ≥ ε} 2 ❉♦ ✤â P({ω : |fn (ω) + gn (ω) − f (ω) − g(ω)| ≥ ε}) 1 ≤ P({ω : |fn (ω) − f (ω)| ≥ ε}) + P({ω : |gn (ω) − g(ω)| ≥ ε}) → 2 ❦❤✐ n → ∞ ❈✉è✐ ❝ò♥❣ t❛ s➩ ①➨t ❜❛ tr÷í♥❣ ❤đ♣✿ ✹✼ ❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû fn → t❤❡♦ ①→❝ s✉➜t✱ g ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦➻✱ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ fn g → t❤❡♦ ①→❝ s✉➜t✳ ❚❤➟t ✈➟②✱ t❛ ✤➦t Bm = {ω : |g(ω)| < m} ✈ỵ✐ ♠é✐ ♠ ∈ N Bm = Ω✳ ❑❤✐ ✤â t❛ ❝â P(Bm ) ↑ P(Ω) = ❉♦ ✤â ❚❛ t❤➜② Bm ⊆ Bm+1 ✈➔ m P({ω : |g(ω)| ≥ m}) = P((Bm )c ) → ❦❤✐ ♠ → ∞ ◆❤÷♥❣ t ợ > trữợ t õ {ω : |fn (ω)g(ω)| ≥ ε} ⊆ {ω : |fn (ω)| ≥ ε } ∪ {ω : |g(ω)| ≥ m}, m ❞♦ ✤â P({ω : |fn (ω)g(ω)| ≥ ε}) ≤ P({ω : |fn (ω)| ≥ ε }) + P({ω : |g(ω)| ≥ m}) m ✭✹✳✹✮ ✈ỵ✐ m ∈ N ❱ỵ✐ > trữợ ố m s P({ω : |g(ω)| ≥ m}) < δ ❑❤✐ õ tỗ t N s ợ n > N t❛ ❝â P({ω : |fn (ω)| ≥ ε }) < δ m ❚ø ✤â t❤❡♦ (4.4) t❛ ❝â P({ω : |fn (ω)g(ω)| ≥ ε}) < δ, ✈ỵ✐ ♥ > N, tù❝ ❧➔ fn g → t❤❡♦ ①→❝ s✉➜t✳ ❚r÷í♥❣ ❤đ♣ ✷✳ ●✐↔ sû fn → ✈➔ gn → t❤❡♦ ①→❝ s✉➙t✳ ❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ fn gn → t❤❡♦ ①→❝ s✉➜t✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ε > ❜➜t ❦➻ t❛ ❝â {ω : |fn (ω)gn (ω)| ≥ ε} ⊆ {ω : |fn (ω)| ≥ ε} ∪ {ω : |gn (ω)| ≤ 1} ❉♦ ✤â P({ω : |fn (ω)gn (ω)| ≥ ε}) ≤ P({ω : |fn (ω)| ≥ ε}) + P({ω : |gn (ω)| ≤ 1}) → ❦❤✐ n → ∞✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä fn gn → t❤❡♦ ①→❝ s✉➜t✳ ❚r÷í♥❣ ❤đ♣ ✸✳ ●✐↔ sû fn → f ✈➔ gn → g t❤❡♦ ①→❝ s✉➜t✳ ❑❤✐ ✤â t❛ ✈✐➳t fn gn − f g = (fn − f )(gn − g) + (fn − f )g + f (gn − g) ✹✽ ❚❤❡♦ ❣✐↔ t❤✐➳t✱ fn → ✈➔ gn → t❤❡♦ ①→❝ s✉➜t ♥➯♥ t❛ ❝â (fn − f ) → ✈➔ (gn − g) → t❤❡♦ ①→❝ s✉➜t✳ ❙û ❞ö♥❣ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â ð tr➯♥ t❛ ♥❤➟♥ t❤➜② (fn −f )(gn −g) → t❤❡♦ ①→❝ s✉➜t✱ (fn − f )g → t❤❡♦ ①→❝ s✉➜t ✈➔ f (gn − g) → t❤❡♦ ①→❝ s✉➜t✳ ❉♦ ✤â (fn gn − f g) → t❤❡♦ ①→❝ s✉➜t ❤❛② fn gn → f g t❤❡♦ ①→❝ s✉➜t✳ ▼➺♥❤ ✤➲ ✹✳✹✳✶✵✳ ◆➳✉ fn −−−→ f t❤➻ fn → f t❤❡♦ ①→❝ s✉➜t✳ ❤✳❝✳❝ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ > > trữợ t ✤➦t gn = |fn − f | ✈➔ A = {ω : gn (ω) → ❦❤✐ ♥ → ∞} ❉♦ (fn − f ) → t❤❡♦ ①→❝ s✉➜t ♥➯♥ P({ω : |fn (ω) − f (ω)| → ❦❤✐ ❱ỵ✐ m ∈ N✱ t❛ ❞➦t Am = {ω : gn (ω) < Akm = {ω : gj (ω) < m }, n ✈ỵ✐ ♠å✐ m ♥ → ∞}) = P(A) = ✤õ ❧ỵ♥✳ ❑❤✐ ✤â Am = Akm ✈ỵ✐ k ❥ ≥ k} = {ω : gj (ω) < j>k } m ❚❛ ❝â A ⊆ Am ❞♦ ✤â P(Am ) = 1✳ ❍ì♥ ♥ú❛ t❛ ❝â Akm ⊆ Ak+1 m ✱ ❞♦ ✤â P(Akm ) ↑ P Akm = P(Am ) = ❦❤✐ ❦ → ∞ k ❈è ✤à♥❤ m ∈ N s❛♦ ❝❤♦ m < ε✳ ❑❤✐ ✤â t❛ ❝â P(Akm ) > − δ ✈ỵ✐ k ✤õ ❧ỵ♥✱ ♥❣❤➽❛ ❧➔ P({ω : gj (ω) < , ∀j > k}) > − δ ✈ỵ✐ m ❦ ✤õ ❧ỵ♥ ❉♦ ✤â t❛ ❝â {ω : gj (ω) < 1 , ∀j > k} ⊇ {ω : gj (ω) < , ∀j ≥ k}), m m tø ✤â P({ω : gj (ω) < 1 , ∀j > k}) ≥ P({ω : gj (ω) < , ∀j ≥ k}) > − δ m m ✈ỵ✐ k ✤õ ❧ỵ♥✳ ❚ø ✤â t❛ ❝â P({ω : gj (ω) ≥ ε}) ≤ P({ω : gj (ω) < ✈ỵ✐ k ✤õ ❧ỵ♥ ✈➔ j > k ✳ ✹✾ }) < δ m ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿ ✲ ❚r➻♥❤ ❜➔②✱ ❤➺ t❤è♥❣ ❤â❛ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ✤ë ✤♦ ✈➔ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ♥❤÷✿ ✣↕✐ sè ✈➔ σ ✲ ✤↕✐ sè✱ ✤ë ✤♦ tr➯♥ ✤↕✐ sè t➟♣ ❤ñ♣✱ ♠ð rë♥❣ ✤ë ✤♦✱ ✤ë ✤♦ tr♦♥❣ Rk ✱ ❤➔♠ ✤♦ ✤÷đ❝✱ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡✳ ✲ ❱➲ σ ✲ ✤↕✐ sè ❇♦r❡❧ tr♦♥❣ R ✈➔ ❤➔♠ sè ❇♦r❡❧✳ ữợ ự ổ st ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ✲ ◆❣❤✐➯♥ ❝ù✉ ✈➔ ♣❤→t tr✐➸♥ ❤ì♥ ỳ ỵ tt t tr ổ ✤♦ ❤ú✉ ❤↕♥ (X, F, µ) ❜➜t ❦➻✳ ✲ ◆❣❤✐➯♥ ❝ù✉ ✈➲ ❦ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ✲ ◆❣❤✐➯♥ ❝ù✉ ✈➲ sü ✤ë❝ ❧➟♣ ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ✈ỵ✐ ❤❛✐ ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✿ ❤ë✐ tö ❤➛✉ tử t st ữợ t tr ❝õ❛ ✤➲ t➔✐✿ ◗✉❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ✤➲ t➔✐ ❝ô♥❣ t r ỳ ữợ ự t t ữ ự t số ợ ỵ ợ ❤↕♥ tr✉♥❣ t➙♠ ❝ò♥❣ ✈ỵ✐ ♥❤ú♥❣ ù♥❣ ❞ư♥❣ q✉❛♥ trå♥❣ ỵ tt st tr tỹ t r q tr➻♥❤ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥✱ ♠➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ s♦♥❣ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦✱ ✤â♥❣ õ ỵ t ổ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ P❤↕♠ ❱➠♥ ❑✐➲✉ ✭✷✵✵✹✮✱ ●✐→♦ tr➻♥❤ ❳→❝ s✉➜t ✈➔ ❚❤è♥❣ ❦➯✱ ◆①❜ ●❉✳ ❬✷❪ ũ ỵ tt s✉➜t ✈➔ ❝→❝ ù♥❣ ❞ö♥❣✱ ◆①❜ ●❉✳ ❬✸❪ P❤↕♠ ▼✐♥❤ ❚❤æ♥❣ ✭✷✵✵✼✮✱ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ✲ ✣ë ✤♦ ✲ ❚➼❝❤ ♣❤➙♥✱ ◆①❜ ●❉✳ ❬✹❪ ◆❣✉②➵♥ ❉✉② ❚✐➳♥✱ ❱ô ❱✐➳t ❨➯♥ ỵ tt st ❲✐❧❞❡✱ ▼❡❛s✉r❡✱ ■♥t❡❣r❛t✐♦♥ ❛♥❞ Pr♦❜❛❜✐❧✐t②✱ ❑✐♥❣✬ s ❈♦❧❧❡❣❡ ▲♦♥❞♦♥✳ ✺✶ ... x ∈ X trữợ t t n0 ợ s f (x) < n0 ✱ ♥❣❤➽❛ ❧➔ x ∈ / Fn ✈ỵ✐ n ≥ n0 ✳ ❉♦ ✤â ✈ỵ✐ ♠å✐ n n0 tỗ t số i ✤â s❛♦ ❝❤♦ x ∈ En,i ❉♦ ✤â t❛ ❝â ≤ f (x) − sn (x) < i i−1 − n = n → ❦❤✐ n → ∞, ∀x ∈ X n 2