▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✿▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❳→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ❝→❝ ❜✐➳♥ ❝è ✤ë❝ ❧➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❈❤÷ì♥❣ ✷✿ ▼ët sè ✈➜♥ ✤➲ ✈➲ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➔ ù♥❣ ❞ö♥❣ ✷✳✶ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➔ ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈ỵ✐ ①→❝ s✉➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ ▼ð rë♥❣ ❝õ❛ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶ ▲❮■ ◆➶■ tứ t ỵ tt s✉➜t ♥❣❤✐➯♥ ❝ù✉ q✉② ❧✉➟t ❝õ❛ ❝→❝ ❤✐➺♥ t÷đ♥❣ ♥❣➝✉ ỹ t tỹ ỵ tt s✉➜t✱ t❤è♥❣ ❦➯ t♦→♥ ①➙② ❞ü♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ r❛ q✉②➳t ✤à♥❤ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ t❤æ♥❣ t✐♥ ❦❤æ♥❣ ✤➛② ✤õ✳ ❍ì♥ ✸✵✵ ♥➠♠ ♣❤→t tr✐➸♥✱ ✤➳♥ ♥❛② ♥ë✐ ❞✉♥❣ ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ①→❝ s✉➜t ✈➔ t❤è♥❣ ❦➯ t♦→♥ r➜t ♣❤♦♥❣ ♣❤ó✱ ✤÷đ❝ ù♥❣ ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ỹ tỹ r ỵ t❤✉②➳t ①→❝ s✉➜t✱ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❜ê ✤➲ ❝â ✈❛✐ trá q✉❛♥ trå♥❣✳ ❈ị♥❣ ✈ỵ✐ sü t tr ỵ tt st rt ✤➣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝✳ ❍✐➺♥ ♥❛② ✤➣ ❝â ♥❤✐➲✉ ♥❣❤✐➯♥ ❝ù✉ ❞➔♥❤ ❝❤♦ ✤✐➲✉ ❦✐➺♥ t❤ù ❤❛✐ ❝õ❛ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ tr♦♥❣ ♥é ❧ü❝ t❤❛② t❤➳ ✤✐➲✉ ❦✐➺♥ ✤ë❝ ❧➟♣ ❝õ❛ ❞➣② ❜✐➳♥ ❝è {An, n ≥ 1} ❜ð✐ ✤✐➲✉ ❦✐➺♥ ②➳✉ ỡ t roăs e ✤➣ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤ë❝ ❧➟♣ tr♦♥❣ ♣❤➛♥ ❤❛✐ ❝õ❛ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ❝â t❤➸ ✤÷đ❝ t❤❛② t❤➳ ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✤ë❝ ❧➟♣ ✤æ✐ ♠ët ❝õ❛ ố {An, n 1} roăs e ❝ô♥❣ ✤➣ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ✤ë❝ ❧➟♣ ✤æ✐ ♠ët ❝õ❛ {An, n ≥ 1} ❝â t❤➸ t❤❛② t❤➳ ❜➡♥❣ ✤✐➲✉ ❦✐➺♥ P (Ak Aj ) ≤ P (Ak )P (Aj ) ✈ỵ✐ ∀k, j : k = j ✳ ◆➠♠ ✶✾✻✶✱ ❦➳t q✉↔ ♥➔② ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ ✤ë❝ ❧➟♣ ❜ð✐ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❑♦❝❤❡♥✱ ❙t♦♥❡ ✈➔ ❙♣✐t③❡r✳ ✣➳♥ ♥➠♠ ✶✾✻✸✱ ▲❛♠♣❡rt✐ ✤➣ ①➙② ❞ü♥❣ t❤➔♥❤ ♠➺♥❤ ✤➲✿ ●✐↔ sû {An, n ≥ 1} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ❝è✱ ♥➳✉ ∞n=1 P (An) = ∞ ✈➔ P (Ak Aj ) ≤ CP (Ak )P (Aj ) ✈ỵ✐ ♠å✐ k, j > N ✭✈ỵ✐ ❈✱ ◆ ❧➔ ❤➡♥❣ sè✮ t❤➻ P (lim sup An ) > ữ r ỵ tờ qt ỡ ✈➔ ❝❤➼♥❤ ①→❝ ❤ì♥ ✈➲ ❝→❝ ❦➳t q✉↔ ♥â✐ tr➯♥✱ tr➯♥ ❝ì sð ❜➔✐ ❜→♦ ❆ ◆♦t❡ ❖♥ ❚❤❡ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ▲❡♠♠❛ ❬✷❪✱ ❝ị♥❣ ✷ ✈ỵ✐ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ữợ sỹ ữợ P ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣✱ ❝❤ó♥❣ tỉ✐ ✤➣ t✐➳♣ ❝➟♥ ✈ỵ✐ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ✧▼ët sè ✈➜♥ ✤➲ ✈➲ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➔ ù♥❣ ❞ư♥❣✧✳ ❱ỵ✐ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ tæ✐ s➩ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➔ ♥➯✉ r❛ ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❇ê ✤➲ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ❝õ❛ ❞➣②✳ ❇➯♥ ❝↕♥❤ ✤â✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➣ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ð rë♥❣ rt ố ỗ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì sð ❧✐➯♥ q✉❛♥ ❝❤➼♥❤ ✤➳♥ ♥ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ s❛✉✳ ❈ư t❤➸✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ♥❤÷ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t❀ ①→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ❝→❝ ❜✐➳♥ ❝è ✤ë❝ ❧➟♣❀ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❦ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè ✈➜♥ ✤➲ ✈➲ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➔ ù♥❣ ❞ö♥❣ ✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ s➩ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ữỡ ữỡ ỗ ử ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❇ê ✤➲ ❇♦r❡❧✲ ❈❛♥t❡❧❧✐✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ✈➔ ♥➯✉ r❛ ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❇ê ✤➲ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ❝õ❛ ❞➣②✳ ▼ư❝ ✷✳✷ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈ỵ✐ ①→❝ s✉➜t ♥❤➡♠ ♣❤ư❝ ✈ư ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ♠ð rë♥❣ ❝õ❛ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐✳ ▼ö❝ ✷✳✸ ❝❤ó♥❣ tỉ✐ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ♠ð rë♥❣✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ữợ sỹ ữợ t t ❝õ❛ ❚❤➛② ❣✐→♦✱ P●❙✳ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t ỗ tớ t ỷ ì♥ tỵ✐ ❇❛♥ ❈❤õ ♥❤✐➯♠ ❦❤♦❛ ❚♦→♥✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ❈✉è✐ ❝ò♥❣✱ t→❝ ❣✐↔ ①✐♥ ỷ ỡ tợ ữớ t t➜t ✸ ❝↔ ❜↕♥ ❜➧❀ ✤➦❝ ❜✐➺t ❧➔ t➟♣ t❤➸ ❧ỵ♣ ❈❛♦ ❤å❝ ✶✽❳❙❚❑ ❚♦→♥ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ▼➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ✈➻ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ ữủ ỳ ỳ ỵ õ õ qỵ t ổ ❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✷ ❚→❝ ❣✐↔ ✹ ❈❍×❒◆● ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✶✳✶✳✶ ✣↕✐ sè ✈➔ σ✲✤↕✐ sè ●✐↔ sû Ω = ∅ ✈➔ P(Ω) ❧➔ ❤å t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ Ω✳ ▼é✐ ❤å C ⊂ P(Ω) s➩ ✤÷đ❝ ❣å✐ ởt ợ ợ A P() ữủ ❣å✐ ❧➔ ♠ët ✤↕✐ sè ♥➳✉ ✭✐✮ Ω ∈ A✱ ✭✐✐✮ A ∈ A =⇒ Ac = Ω\A ∈ A ❀ ✭✐✐✐✮ A, B ∈ A ⇒ A B ∈ A ợ F P() ữủ ởt sè ♥➳✉ ✭✐✮ Ω ∈ F ✱ ✭✐✐✮ A ∈ F =⇒ Ac = Ω\A ∈ F ❀ ∞ ✭✐✐✐✮ An ∈ F(∀n = 1, 2, 3, ) ⇒ An ∈ F ✳ n=1 ❚ø ❝→❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ ❝â t❤➸ s✉② r❛ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙② ✭✶✮ ❚r♦♥❣ ✤✐➲✉ ❦✐➺♥ ✭✐✐✐✮ ❝õ❛ ∞❝→❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❝â t❤➸ t❤❛② ✤✐➲✉ ❦✐➺♥ ∞ A B ∈ A ❜ð✐ A B ∈ A ✈➔ An ∈ F ❜ð✐ An ∈ F ✳ n=1 n=1 ✭✷✮ ◆➳✉ A ❧➔ ✤↕✐ sè✱ A, B ∈ A t❤➻ A \ B ∈ A✳ ✭✸✮ ◆➳✉ F ❧➔ σ✲✤↕✐ sè t❤➻ F ❧➔ ✤↕✐ sè✳ ✭✹✮ ●✐❛♦ ❝õ❛ ♠ët ❤å ❜➜t ❦➻ ❝→❝ ✤↕✐ sè ✭σ✲✤↕✐ sè✮ ❧➔ ♠ët ✤↕✐ sè ✭σ✲✤↕✐ sè✮✳ ✶✳✶✳✷ σ✲✤↕✐ sè s✐♥❤ ❜ð✐ ♠ët ❧ỵ♣ ✈➔ σ✲✤↕✐ sè ❇♦r❡❧ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû C ∈ P(Ω)✳ ❑❤✐ ✤â ✤↕✐ sè ✭σ✲✤↕✐ sè ✮ ❜➨ ♥❤➜t ❝❤ù❛ ✤÷đ❝ ❣å✐ ❧➔ ✤↕✐ sè ✭σ✲✤↕✐ sè✮ s✐♥❤ ❜ð✐ C ✱ ❦➼ ❤✐➺✉ ❧➔ A(C) (σ(C))✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû (X, T ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ ❦❤✐ ✤â σ✲✤↕✐ sè s✐♥❤ ❜ð✐ T ✤÷đ❝ ❣å✐ ❧➔ σ ✲✤↕✐ sè ❇♦r❡❧ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ B(X)✳ C ✶✳✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ✤♦ ✈➔ ✤ë ✤♦ ①→❝ s✉➜t ●✐↔ sû Ω ❧➔ ♠ët t➟♣ tũ ỵ rộ F ởt số t➟♣ ❝♦♥ ❝õ❛ Ω✳ ❑❤✐ ✤â✱ ❝➦♣ (Ω, F) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤♦✳ ●✐↔ sû (Ω, F)❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤♦✳ ▼ët ⑩♥❤ ①↕ P : F → R ✤÷đ❝ ❣å✐ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t tr➯♥ F ♥➳✉ ✭✐✮ P(A) ✈ỵ✐ ∀A ∈ F ✭t➼♥❤ ❦❤æ♥❣ ➙♠✮❀ ✭✐✐✮ P(Ω) = 1✭t➼♥❤ ❝❤✉➞♥ ❤â❛✮❀ ✭✐✐✐✮ ◆➳✉ An ∈ F(∀n = 1, 2, 3, ), Ai Aj = AiAj = ∅(i = j) t❤➻ ∞ P( ∞ An ) = n=1 P(An ) n=1 ✭t➼♥❤ ❝ë♥❣ t➼♥❤ ✤➳♠ ✤÷đ❝✮✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✭✐✮✲✭✐✐✐✮ ✤÷đ❝ ❣å✐ ❧➔ ❍➺ t✐➯♥ ✤➲ ❑♦❧♠♦❣♦r♦✈ ✈➲ ①→❝ s✉➜t✳ ❇ë ❜❛ (Ω, F, P)✤÷đ❝ ❣å✐ ❧➔ ❑❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳ ❚➟♣ Ω ✤÷đ❝ ❣å✐ ❧➔ ❑❤ỉ♥❣ ❣✐❛♥ ❜✐➳♥ ❝è ❝➜♣✳ σ ✲✤↕✐ sè F ✤÷đ❝ ❣å✐ ❧➔ σ ✲✤↕✐ sè ❝→❝ ❜✐➳♥ ❝è✳ ▼é✐ A ∈ F ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❜✐➳♥ ❝è ✳ ❇✐➳♥ ❝è Ω ∈ F ✤÷đ❝ ❣å✐ ❧➔ ❜✐➳♥ ❝è ❝❤➢❝ ❝❤➢♥✳ ❇✐➳♥ ❝è ∅ ∈ F ✤÷đ❝ ❣å✐ ❧➔ ❜✐➳♥ ❝è ❦❤æ♥❣ t❤➸ ❝â✳ ❇✐➳♥ ❝è A = Ω \ A ✤÷đ❝ ❣å✐ ❧➔ ❜✐➳♥ ❝è ✤è✐ ❧➟♣ ❝õ❛ ❜✐➳♥ ❝è ❆✳ ◆➳✉ A B = AB = ∅ t❤➻ ❆✱ ❇ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❜✐➳♥ ❝è ①✉♥❣ ❦❤➢❝✳ ✻ ✶✳✶✳✹ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ①→❝ s✉➜t ●✐↔ sû ❆✱ ❇✱ ❈✱ ✳✳✳ ❧➔ ♥❤ú♥❣ ❜✐➳♥ ❝è✳ ❑❤✐ ✤â✱ ①→❝ s✉➜t ❝õ❛ ❝❤ó♥❣ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ✶✳ P(∅) = 0✳ ❚❤➟t ✈➟②✱ ✤➦t A∞1 = Ω; An∞= ∅(∀n > 1)✳ ❑❤✐ ✤â✱ sû ❞ö♥❣ ✭✐✐✐✮ t❛ ✤÷đ❝ ∞ P(An )✳ P(An ) = P(Ω) + = P(Ω) = P( An ) = n=2 n=1 n=1 ❙✉② r❛ ∞ P(An ) = 0✳ n=2 ✣✐➲✉ ♥➔②✱ ❝ị♥❣ ✈ỵ✐ ✭✐✮ ❝❤♦ t❛ P(∅) = P(An ) = 0(∀n > 1)✳ ✷✳ ◆➳✉ AB = ∅ t❤➻ P(A ∪ B) = P(A) + P(B) ✳ ❚➼♥❤ ❝❤➜t ♥➔② ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ t✐➯♥ ✤➲ ✈➲ t➼♥❤ ❝ë♥❣ t➼♥❤ ✤➳♠ ✤÷đ❝ ❝õ❛ ✤ë ✤♦ ①→❝ s✉➜t ✈➔ t➼♥❤ ❝❤➜t ✶✳ ✸✳ P(A) = − P(A)✳ ❚❤➟t ✈➟②✱ t❛ ❝â A ∪ A = Ω, AA = ∅ ❙✉② r❛ = P(Ω) = P(A ∪ A) = P(A) + P(A), ♥➯♥ P(A) = − P(A) ✹✳ ◆➳✉ A ⊂ B t❤➻ P(B \ A) = P(B) − P(A) ✈➔ ❞♦ ✤â P(A) ≤ P(B) ✳ ❚❤➟t ✈➟② B = A ∪ (B \ A), A(B \ A) = ∅ ♥➯♥ P(B) = P(A) + P(B \ A) ≥ P(A) ✼ ❙✉② r❛ P(B \ A) = P(B) − P(A) ✺✳ P(A ∪ B) = P(A) + P(B) − P(AB)✳ ❚❤➟t ✈➟②✱ t❛ ❝â A ∪ B = A ∪ (B \ AB); A(B \ AB) = ∅; AB ⊂ B ❙✉② r❛ P(A ∪ B) = P(A) + P(B \ AB) = P(A) + P − P(AB) ✻✳ P( n n + (−1) P(Ak ) − Ak ) = k=1 n−1 k=1 P(Ak Al Am ) − P(Ak Ai ) + 1≤k 0✱ P(B) > 0✱ t❤➻ A, B ✤ë❝ ❧➟♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ P(A/B) = P(A) ❤♦➦❝ P(B/A) = P(B) ✷✳ ❍❛✐ ❜✐➳♥ ❝è ❆ ✈➔ ❇ ✤ë❝ ❧➟♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➯♥ s❛✉ t❤ä❛ ♠➣♥ ✭✐✮ A, B ✤ë❝ ❧➟♣❀ ✭✐✐✮ A, B ✤ë❝ ❧➟♣❀ ✶✵ ✶✳✸✳✸ ❑ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû ❳ ✿ (Ω, F, P) → (R, B(R)) ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ❝õ❛ ❳ t❤❡♦ ✤ë ✤♦ P ✭♥➳✉ tỗ t ữủ ý ỵ EX EX = XdP ✣➦❝ ❜✐➺t✱ ♥➳✉ ❳ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ X = n IAi i=1 t❤➻ n EX = P(Ai ) i=1 ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦ý ✈å♥❣✳ ✶✳ ◆➳✉ X ≥ t❤➻ EX ≥ ✷✳ ◆➳✉ X = C t❤➻ EX = C ✸✳ tỗ t EX t ợ C R t õ E(CX) = C EX tỗ t EX ✈➔ EY t❤➻ E(X ± Y ) = EX ± EY ✶✳✸✳✹ ❈→❝ ❞↕♥❣ ❤ë✐ tö ✣à♥❤ ♥❣❤➽❛✳ ❚❛ ♥â✐ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ (Xn, n ≥ 1) ❤ë✐ tö ✤➳♥ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❳ ✭❦❤✐ n → ∞✮ • ❍➛✉ ❝❤➢❝ ❝❤➢♥ ♥➳✉ P( lim | Xn − X |= 0) = n ỵ Xn h.c.c X ã st ợ ε > t❤➻ lim P(| Xn − X |> ) = n P ỵ Xn X ã ợ > t❤➻ ∞ P(| Xn − X |> ε) < ∞ n=1 c ỵ Xn X ì ✷ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ❱➋ ❇✃ ✣➋ ❇❖❘❊▲✲❈❆◆❚❊▲▲■ ❱⑨ Ù◆● ❉Ư◆● ✷✳✶ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➔ ù♥❣ ❞ư♥❣ ✷✳✶✳✶ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ●✐↔ sû (A∞n, n ≥ 1) ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ❝è✳ ❑❤✐ ✤â ✭✐✮ ◆➳✉ P(An) < ∞ t❤➻ P(lim sup An) = 0✳ n=1 ∞ ✭✐✐✮ ◆➳✉ P(An) = ∞ ✈➔ (An, n ≥ 1) ✤ë❝ ❧➟♣ t❤➻ P(lim sup An) = 1✳ n=1 ❚r♦♥❣ ✤â ∞ ∞ Ak lim sup An = n=1 k=n ❈❤ù♥❣ ♠✐♥❤✳ ∞ ✭✐✮ ●✐↔ sû n=1 P(An ) < ∞✳ ∞ ❑❤✐ ✤â ∞ P(lim sup An ) = P( ∞ ∞ Ak ) ≤ lim Ak ) = lim P( n→∞ n=1 k=n k=n n→∞ P(Ak ) = k=n rữợ t t r ≤ x ≤ t❤➻ − x ≤ e−x ✳ ∞ ●✐↔ sû P(An) = ∞✳ ❑❤✐ ✤â✱ ✈➻ ❞➣② (An, n ≥ 1) ✤ë❝ ❧➟♣ ♥➯♥ ❞➣② n=1 (An , n ≥ 1) ❝ô♥❣ ✤ë❝ ❧➟♣✳ ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ n = 1, 2, ✈➔ ♠å✐ m > n✱ t❛ ❝â m − P( k=n m A k ) = P( k=n m Ak ) = P( k=n ∞ m (1 − P(Ak )) ≤ e Ak ) = k=n − P(Ak ) k=n ❙✉② r❛ ∞ ≤ − P( Ak ) = lim (1 − P( m→∞ k=n ❉♦ ✤â P( ∞ m ∞ Ak ) = k=n Ak )) ≤ lim e − P(Ak ) k=n m→∞ k=n ✈ỵ✐ ♠å✐ n = 1, 2, ✳✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ ∞ Ak ) = P(lim sup An ) = lim P( n→∞ k=n ❇ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✶✳✷ ▼➺♥❤ ✤➲ ◆➳✉ (Xn, n ≥ 1) ❧➔ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ c t❤➻ Xn −h.c.c −→ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Xn → − C✳ ❈❤ù♥❣ ♠✐♥❤✳ c h.c.c ●✐↔ sû Xn → − C t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ Xn −−→ C c ❚❤➟t ✈➟②✱ ❞♦ Xn → − C ♥➯♥ ∞ P (| Xm − C |> ) → m=n ♠➔ t❛ ❧↕✐ ❝â ∞ (| Xm − C |> ) P ❙✉② r❛ ∞ ≤ m=n P (| Xm − C |> ) m=n ∞ (| Xm − C |> ) lim P n→∞ =0 m=n ❉♦ ✤â lim P sup | Xm − C |> n→∞ m≥n ❙✉② r❛ Xn −h.c.c −→ C ✶✺ = =0 c ●✐↔ sû ♥❣÷đ❝ ❧↕✐ Xn −h.c.c −→ C t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ Xn → − C ❚❤➟t ✈➟②✱ t❛ ✤➦t An = (| Xn − C |> ) ❑❤✐ ✤â✱ (An) ❧➔ ❞➣② ❜✐➳♥ ❝è ✤ë❝ ❧➟♣✳ ❚❛ ❝â ∞ ∞ ∞ ∞ (| Xm − C |> ) Ak = lim sup An = n=1 k=n n=1 k=n ❙✉② r❛ lim sup An ⊂ (Xn ▼➦t ❦❤→❝✱ ❞♦ Xn −h.c.c −→ C ♥➯♥ P(Xn ❙✉② r❛ C) C) = P(lim sup An ) = ❉♦ ❞➣② (An) ✤ë❝ ❧➟♣ ♥➯♥ →♣ ❞ö♥❣ ❇ê ✤➲ ❇♦r❡❧✲❈❛♥t❡❧❧✐ t❛ s✉② r❛✿ ∞ P(An ) < ∞ n=1 ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ >0 t❤➻ ∞ P(| Xm − C |> ) < ∞ n=1 c − C✳ ❙✉② r❛ Xn → ❱➟② h.c.c c − C Xn −−→ C ⇔ Xn → ✷✳✶✳✸ ▼➺♥❤ ✤➲ ◆➳✉ ❞➣② ỡ t st t tỗ t (Xn ; k ≥ 1) ⊂ (Xn , n ≥ 1) s❛♦ ❝❤♦ (Xn ; k ≥ 1) ❤ë✐ tö ❤✳ ❝✳ ❝✳ (Xn , n ≥ 1) k k ✶✻ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ (Xn, n ≥ 1) ❝ì ❜↔♥ t❤❡♦ ①→❝ s✉➜t ♥➯♥ ✈ỵ✐ ♠å✐ ε > lim P(| Xm − Xn |> ε) = n→∞ ❉♦ õ ợ = 12 tỗ t n1 s ✈ỵ✐ ♠å✐ m, n ≥ n1 t❤➻ 1 P(| Xm − Xn |> ) < 2 ❱ỵ✐ = 21 tỗ t n2 > n1 s ✈ỵ✐ ♠å✐ m, n ≥ n2 t❤➻ P(| Xm − Xn |> 1 ) < 22 22 ❱ỵ✐ k = 21 tỗ t nk > nk s ❝❤♦ ✈ỵ✐ ♠å✐ m, n ≥ nk t❤➻ k P(| Xm − Xn |> ❙✉② r❛ (Xn ) t❤ä❛ ♠➣♥ P(| Xn +1 − Xn ✣➦t k k 1 ) < 2k 2k k |> Ak = (| Xnk +1 − Xnk |> 2k ) < 2k ✳ ), (k ≥ 1) 2k ❑❤✐ ✤â ∞ ∞ P(Ak ) ≤ k=1 k=1 0)E(X ) ❈❤ù♥❣ ♠✐♥❤ ❚❛ ❝â EX = E(XI(X=0) + XI(X>0) ) = E(XI(X=0) ) + E(XI(X>0) ) = E(XI(X>0) ) ✶✾ ❇➙② ❣✐í →♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲ ❇✉♥❤✐❛❦♦✇s❦✐ t❛ ✤÷đ❝ (EX)2 = E(XI(X>0) ) ≤ E(X )E(I(X>0) )2 = E(I(X>0) )2 E(X ) = P(X > 0)E(X ) ự ỵ ✷✳✷✳✷✿ ❳→❝ ✤à♥❤ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ Xk (ω), ω ∈ Ω s❛♦ ❝❤♦ ✿ Xk (ω) = IAk (ω) = ω ∈ Ak , ω ∈ Ak ❚❛ ❝â P(Ak Aj ) = E(X1 + X2 + + XN )2 − E(X12 + X22 + + XN2 ) 1≤k 0)E(X1 + X2 + + XN )2 ✭✷✳✸✮ ❚ø E(Xk ) = E(Xk2) = P(Ak ) ✈➔ P(X1 + X2 + + XN > 0) = P( Ak )✳ k=1 ❑➳t ❤đ♣ ✈ỵ✐(2.1), (2.2) ✈➔ (2.3) t❛ s✉② r❛ N −1 N P(Ak Aj ) ≥ P( 1≤k 0✱ t❤➻ N P Ak k=n ự ỵ t tø ❜➜t ✤➥♥❣ t❤ù❝ (2.1) t❛ s✉② r❛ −1 N P( N Ak ) P(Ak ) k=1 N ≤2 P(Ak Aj ) + 1≤k