Luận văn sư phạm Về định lí Birkhoff và một số vấn đề liên quan trong lí thuyết Ergodic

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ▲❮■ ❈❷▼ ❒◆ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü ❣✐ó♣ ✤ï ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ tê ●✐↔✐ t➼❝❤✱ ❝→❝ t❤➛② ❣✐→♦ ❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛ t♦→♥✱ ❝→❝ t❤➛② ❣✐→♦ ❝ỉ ❣✐→♦ tr♦♥❣ tr÷í♥❣ ✣❍❙P ❍➔ ◆ë✐ ✷ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ tỵ✐ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ ❧➛♥ ✤➛✉ t✐➯♥ ❧➔♠ q✉❡♥ ✈ỵ✐ ❝ỉ♥❣ t→❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱ ❤ì♥ ♥ú❛ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳✱ ♠➦❝ ❞ị r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ ❦➼♥❤ ữủ sỹ õ õ ỵ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❍➔ ◆ë✐✱ ♥❣➔② ✶✵ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✸ ❙✐♥❤ ✈✐➯♥ ❍♦➔♥❣ ❚❤à ▲✐➯♥ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✶ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ▲❮■ ❈❆▼ ✣❖❆◆ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ ❝â t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜ð✐ sü ❝è ❣➢♥❣ ♥é ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥ ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✱ ❜➯♥ ❝↕♥❤ ✤â ❡♠ ♥❤➟♥ ữủ sỹ ữợ t t ❚↕ ◆❣å❝ ❚r➼ ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ t♦→♥✳ ❊♠ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❍➔ ◆ë✐✱ ♥❣➔② ✶✵ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✸ ❙✐♥❤ ✈✐➯♥ ❍♦➔♥❣ ❚❤à ▲✐➯♥ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✷ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ▼ö❝ ❧ö❝ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✶✳✶ ●✐ỵ✐ t❤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ❈→❝ ✈➼ ❞ö ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ✶✳✸ ❚➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ Lp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸ ❈→❝ ✤à♥❤ ❧➼ ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ✣ë ✤♦ ❜➜t ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❊❘●❖❉■❈ ❱⑨ ✣➚◆❍ ▲➑ ❇■❘❑❍❖❋❋ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ❝õ❛ ❊r❣♦❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣➦❝ tr÷♥❣ ❝õ❛ ❊r❣♦❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❈→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ❈→❝ ♣❤➨♣ q✉❛② ♠ët ✤÷í♥❣ trá♥ ✷✳✸✳✷ ⑩♥❤ ①↕ ❦➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✸ ⑩♥❤ ①↕ ❧✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✻ ✽ ✽ ✾ ✶✵ ✶✵ ✶✶ ✶✶ ✶✷ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✹ ✶✻ ✶✻ ✶✼ ✶✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ r ỹ tỗ t ✤♦ ❊r❣♦❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ P❤➨♣ tr✉② t♦→♥ ✈➔ ❊r❣♦❞✐❝ ✤ì♥ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✶ ✣à♥❤ ❧➼ ♣❤➨♣ tr✉② t♦→♥ ❝õ❛ P♦✐♥❝❛r❡ ✳ ✳ ✳ ✳ ✷✳✺✳✷ ❊r❣♦❞✐❝ ✤ì♥ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✸ ❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ✣à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✶ ❑➻ ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✷ ✣à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ t❤❡♦ tø♥❣ ✤✐➸♠ ✷✳✼ ❈→❝ ❤➺ q✉↔ ❝õ❛ ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ✳ ✳ ✳ ✳ ✷✳✼✳✶ ❈→❝ ❤➺ q✉↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✷ Ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ▼ët sè ❜➔✐ t➟♣ ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽✳✶ ❇➔✐ t➟♣ ✶✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽✳✷ ❇➔✐ t➟♣ ✷✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽✳✸ ❇➔✐ t➟♣ ✸✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽✳✹ ❇➔✐ t➟♣ ✹✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✶ ✷✶ ✷✷ ✷✹ ✷✹ ✷✹ ✷✻ ✸✷ ✸✷ ✸✹ ✸✽ ✸✽ ✸✾ ✹✶ ✹✶ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ r é ỵ t➔✐ ●✐↔✐ t➼❝❤ ❤➔♠ ❧➔ ♠ët ♥❣➔♥❤ t♦→♥ ❤å❝ ✤÷đ❝ ①➙② ❞ü♥❣ ✤➛✉ t❤➳ ❦➾ ❳❳ ✈➔ ✤➳♥ ♥❛② ✈➝♥ ✤÷đ❝ ①❡♠ ❧➔ ♠ët ♥❣➔♥❤ t♦→♥ ❤å❝ ❝ê ✤✐➸♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♣❤→t tr✐➸♥ ❣✐↔✐ t➼❝❤ ❤➔♠ ✤➣ t➼❝❤ ❧ô② ✤÷đ❝ ♠ët sè ♥ë✐ ❞✉♥❣ ❤➳t sù❝ ♣❤♦♥❣ ♣❤ó✱ ♥❤ú♥❣ ❦➳t q✉↔ ♠➝✉ ♠ü❝✱ tê♥❣ q✉→t ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠ ✤➣ ①➙♠ ♥❤➟♣ ✈➔♦ t➜t ❝↔ ❝→❝ ♥❣➔♥❤ t♦→♥ ❤å❝ ❝â ❧✐➯♥ q✉❛♥ ✈➔ sû ❞ư♥❣ ✤➳♥ ❝ỉ♥❣ ❝ư ❣✐↔✐ t➼❝❤ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✳ ❈❤➼♥❤ ✤✐➲✉ ✤â ✤➣ ♠ð rë♥❣ ❦❤æ♥❣ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❝→❝ ♥❣➔♥❤ t♦→♥ ❤å❝✳ ợ ố ữủ ự t s s ✈➲ ❜ë ♠æ♥ ♥➔②❀ ✤➦❝ ❜✐➺t ❧➔ ✤✐ s➙✉ ✈➔♦ t➻♠ ❤✐➸✉ ✈➲ ❧➼ t❤✉②➳t ❊r❣♦❞✐❝ ✈➔ ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✱ t➻♠ ❤✐➸✉ ✈➲ ✤à♥❤ ❧➼ ❇✐r❦❤♦❢❢ ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â ❣✐ó♣ ❝→❝ ❜↕♥ ✤å❝ ❤✐➸✉ rã ❤ì♥ ✈➲ ❝❤ó♥❣✳ ❱ỵ✐ ❧➼ ❞♦ tr➯♥ ❝ị♥❣ ✈ỵ✐ sü t sỹ ữợ t t➻♥❤ ❝õ❛ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼✱ ❡♠ ✤➣ ❝❤å♥ ✤➲ t ỵ r ởt số q tr ỵ tt r trú õ õ ỗ ữỡ ữỡ ▼ỉt sè ❦✐➳♥ t❤ù❝ ❝ì sð✳ ❈❤÷ì♥❣ ✷✿ ❊r❣♦❞✐❝ ✈➔ r ự ữợ ❧➔♠ q✉❡♥ ✈ỵ✐ ❝ỉ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ✤➦❝ ❜✐➺t ỵ tt r ỵ r ự ỵ tt r ởt số q ỵ r ự õ Pữỡ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✣å❝ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤✱ s♦ s→♥❤✱ tê♥❣ ❤đ♣✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✺ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❈❤÷ì♥❣ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✶✳✶ ●✐ỵ✐ t❤✐➺✉ ✳ ❈❤♦ ❳ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t♦→♥ ❤å❝✳ ❳➨t →♥❤ ①↕ T : X → X ✳ ▲➜② x ∈ X ✈➔ ❧➦♣ ❧↕✐ ù♥❣ ❞ö♥❣ ❝õ❛ →♥❤ ①↕ ❚ ố ợ t ữủ ởt {x, T (x), T (x), T (x), }✳ ✣➙② ❣å✐ ❧➔ q✉ÿ ✤↕♦ ❝õ❛ ①✳ ◆➳✉ T n(x) = x t❤➻ ữủ t ợ ♥✳ ❚❛ ①➨t ❜➔✐ t♦→♥ ♥❤÷ s❛✉✿ ❈❤♦ T : [0, 1] → [0, 1] ✈➔ ❝è ✤à♥❤ ♠ët ✤♦↕♥ [a, b] ⊂ [0, 1], ❝❤♦ x ∈ [0, 1] ❚➛♥ sè ♠➔ ❝→❝ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ rữợ t t t tr÷♥❣ χA ❝õ❛ t➟♣ ❆ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿  χA =   ✻ ,x ∈ A ,x ∈ /A ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❚❤➻ sè ❧➛♥ ♥ ✤✐➸♠ ✤➛✉ t✐➯♥ tr♦♥❣ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ❬❛✱❜❪ ❧➔✿ n−1 χ[a,b] (T j (x)) j=0 ❉♦ ✤â t➾ ❧➺ ❝õ❛ ♥ ✤✐➸♠ ✤➛✉ t✐➯♥ tr♦♥❣ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ❬❛✱❜❪ ❧➔ n n−1 χ[a,b] (T j (x)) j=0 ❉♦ ✤â t➛♥ sè ♠➔ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ❬❛✱❜❪ ❧➔ lim n→∞ n n−1 χ[a,b] (T j (x)) j=0 ▼ët ❦➳t q✉↔ ❦❤→ q✉❛♥ trå♥❣ ✤â ❧➔ ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ s➩ ❝❤➾ ❝❤♦ ❝❤ó♥❣ t❛ r➡♥❣✿ ❑❤✐ ❚ ❧➔ ❊r❣♦❞✐❝ t❤➻ t➛♥ sè tr➯♥ ❜➡♥❣ ✤ë ❞➔✐ ❝õ❛ ✤♦↕♥ ❬❛✱❜❪✳ ❚ù❝ ❧➔ ✭❚r♦♥❣ tr÷í♥❣ ❤đ♣ ❝õ❛ ✤ë ✤♦ ❊r❣♦❞✐❝✮✿ lim n→∞ n n−1 χ[a,b] (T j (x)) = b − a j=0 ✈ỵ✐ x ∈ X ❤✳❦✳♥✳ ▼ët ❝→❝❤ tê♥❣ q✉→t ❝õ❛ ✤à♥❤ ❧➼ t ợ ữủ f t ❦➻ t❤➻ t➛♥ sè ♠➔ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ♠ët t➟♣ ❝♦♥ A ∈ X ❧➔✿ lim n→∞ n n−1 f (T j x) j=0 ❑❤✐ ❚ r ố ợ t ợ ♥➔② ❧➔✿ n→∞ n n−1 f (T j x) = lim f dà j=0 rữợ ự t ỵ ú t ởt số ❦✐➳♥ t❤ù❝✿ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✼ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ✶✳✷✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✿ ▼ët ❧ỵ♣ M ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳ ✤÷đ❝ ❣å✐ ❧➔ ✤↕✐ sè ♥➳✉✿ ✐✳ ∅ ∈ M; ✐✐✳ ◆➳✉ A, B ∈ M t❤➻ A ∪ B ∈ M; ✐✐✐✳ ◆➳✉ A ∈ M t❤➻ Ac ∈ M ✣à♥❤ ♥❣❤➽❛ ✶✳✷✿ ởt ợ t ữủ ❧➔ σ✲✤↕✐ sè ♥➳✉✿ ✐✳ ∅ ∈ β; ✐✐✳ ◆➳✉ E ∈ β t❤➻ ♣❤➛♥ ❜ò ❝õ❛ ♥â X\E ∈ β; ✐✐✐✳ ◆➳✉ En ∈ β ✱ ♥❂✶✱✷✱✸✳ ✳ ✳ ❧➔ ❞➣② ✤➳♠ ✤÷đ❝ ❝→❝ t➟♣ ❤đ♣ tr♦♥❣ β t❤➻ ∞ En ∈ β n=1 ✣à♥❤ ♥❣❤➽❛ ✶✳✸✿ ❈❤♦ ❳ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ❝♦♠♣❛❝t✳ ▼ët t➟♣ ❤đ♣ σ ✲✤↕✐ sè ❇♦r❡❧ β(X) ✤÷đ❝ ①→❝ ✤à♥❤ ❧➔ σ ✲✤↕✐ sè ♥❤ä ♥❤➜t ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳ ♠➔ ❜❛♦ ❤➔♠ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ♠ð ❝õ❛ ❳✳ ❈❤♦ ❳ ❧➔ ♠ët t➟♣ ✈➔ β ❧➔ ♠ët σ ✲✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳✱ t❛ ❝â✿ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✿ ▼ët ❤➔♠ sè µ : β → R + ∪ {∞} ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤ë ✤♦ ♥➳✉✿ ✐✳ µ(∅) = 0; ✐✐✳ ◆➳✉ En ❧➔ ❝→❝ t➟♣ ❤ñ♣ ✤➳♠ ✤÷đ❝✱ ✤ỉ✐ ♠ët ♣❤➙♥ ❜✐➺t tr♦♥❣ β t❤➻✿ ∞ µ( ∞ En ) = n=1 µ(En ) n=1 ❚❛ ❣å✐ (X, β, µ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤ë ✤♦✳ ◆➳✉ µ(X) < ∞ t❤➻ µ ❧➔ ✤ë ✤♦ ❤ú✉ ❤↕♥✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✽ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ◆➳✉ µ(X) = t❤➻ µ st (X, , à) tữỡ ù♥❣ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✿ ▼ët ❞➣② ❝→❝ ✤ë ✤♦ ①→❝ s✉➜t µn ❤ë✐ tư ②➳✉ n ợ ộ f ∈ C(X, R) f dµn → X f dµ X ❦❤✐ n → ∞ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✿ ❚❛ ♥â✐ ♠ët t➼♥❤ ❝❤➜t ✤ó♥❣ ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ ❳ ♥➳✉ t➟♣ ❤đ♣ ❝→❝ ✤✐➸♠ ♠➔ ❦❤ỉ♥❣ ❝â t➼♥❤ ❝❤➜t ✤â ❝â ✤ë ✤♦ ✵✳ ✶✳✷✳✷ ❈→❝ ✈➼ ❞ư ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✤ë ✤♦ ✣ë ✤♦ ▲❡❜❡s❣✉❡ tr➯♥ ❬✵✱✶❪✳ ▲➜② ❳❂❬✵✱✶❪ ✈➔ ❧➜② M ❧➔ ❧ỵ♣ ❝õ❛ ❝→❝ ❤đ♣ ❤ú✉ ❤↕♥ t➜t ❝↔ ❝→❝ ❦❤♦↔♥❣ ❝♦♥ ❝õ❛ ❬✵✱✶❪✳ ❱ỵ✐ ♠é✐ ✤♦↕♥ ❝♦♥ ❬❛✱❜❪✱ ✤à♥❤ ♥❣❤➽❛✿ µ ([a, b]) = b − a ❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡✳ ✣ë ✤♦ ▲❡❜❡s❣✉❡ tr➯♥ R/Z✳ ▲➜② ❳❂R/Z❂❬✵✱✶✮ ♠♦❞ ✶ ✈➔ ❧➜② M ❧➔ ❧ỵ♣ ❝õ❛ ❝→❝ ❤ñ♣ ❤ú✉ ❤↕♥ t➜t ❝↔ ❝→❝ ❦❤♦↔♥❣ ❝♦♥ ❝õ❛ ❬✵✱✶✮✳ ợ ởt ([a, b]) = b − a ❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ tr➯♥ ✤÷í♥❣ trá♥✳ ✣ë ✤♦ ❉✐r❛❝✳ ❈❤♦ ❳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✈➔ β ❧➔ ♠ët σ ✲✤↕✐ sè ❜➜t ❦➻✳ ❈❤♦ x ∈ X ✳ ✣à♥❤ ♥❣❤➽❛ ✤ë ✤♦ δx ❜ð✐✿   δx (A) =  ,x ∈ A ,x ∈ /A ❚❤➻ δx ❧➔ ✤ë ✤♦ ①→❝ s✉➜t✳ ◆â ✤÷đ❝ ❣å✐ ❧➔ ✤ë ✤♦ ❉✐r❛❝ t↕✐ ①✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✾ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✶✳✸ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❚➼❝❤ ♣❤➙♥ ❈❤♦ (X, β, µ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤ë ✤♦✳ ✶✳✸✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✿ ▼ët ❤➔♠ sè f : X → R ❧➔ ✤♦ ✤÷đ❝ ♥➳✉ f −1 [(c, ∞)] ∈ β ✈ỵ✐ ∀c ∈ R ✣à♥❤ ♥❣❤➽❛ ✶✳✽✿ ▼ët ❤➔♠ sè f : X → R ❧➔ ✤ì♥ ❣✐↔♥ tr➯♥ ❳ ♥➳✉ ♥â ❝â t❤➸ ✈✐➳t ♥❤÷ tê ❤đ♣ t✉②➳♥ t➼♥❤ ❝→❝ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ t➟♣ tr♦♥❣ β ✱ ♥❣❤➽❛ ❧➔ r f= χAi i=1 ✈ỵ✐ ∈ R, Ai ∈ β, Ai ✤æ✐ ♠ët ❦❤æ♥❣ ❣✐❛♦ ♥❤❛✉ ✈➔ X = r Ai ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✾✿ ❱ỵ✐ ♠ët ❤➔♠ ✤ì♥ ❣✐↔♥ f : X → R ✱ t❛ ❣å✐ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ f tr➯♥ ❳ ❦➼ ❤✐➺✉ X i=1 f dµ ①→❝ ✤à♥❤ ❜ð✐ r f dµ = X ✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✿ µ(Ai ) i=1 f t tỗ t ởt ❞➣② ❤➔♠ ✤ì♥ ❣✐↔♥ t➠♥❣ fn s❛♦ ❝❤♦ fn ↑ f ❦❤✐ n → ∞✳ ❑❤✐ ✤â t❛ ❣å✐ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ❦❤æ♥❣ ➙♠ f tr➯♥ ❳ ❦➼ ❤✐➺✉ X f dµ ①→❝ ✤à♥❤ ❜ð✐ f dµ = lim X ✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✿ n→∞ fn dµ ◆➳✉ f ❝â ❞➜✉ ❜➜t ❦➻✱ t❛ ✤➦t f = f + − f − ✈ỵ✐ f + = max{f, 0} ≥ ✈➔ f − = max{−f, 0} ≥ t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ f ❜➜t ❦➻ tr➯♥ ❳ ❦➼ ❤✐➺✉ X f dµ ①→❝ ✤à♥❤ ❜ð✐ f dµ = f + dµ − f − dµ X ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✶✵ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❚❤❛② f, α ✈➔ γ ❜ð✐ −f, −γ ✈➔ −α ✈➔ sû ❞ö♥❣ ❣✐↔ t❤✐➳t −f ∗ = −f∗ ✈➔ −f∗ = −f ∗ ✱ t ữủ dà à(E, ) E, õ à(E, ) ≤ γµ(Eα,γ ) ❱➻ γ < α ♥➯♥ γµ(Eα,γ ) = 0✳ ❉♦ ✤â f ∗ = f∗ µ − h.k.n ✈➔ lim n→∞ n n−1 f (T j x) = f ∗ (x) µ − h.k.n (1) j=0 •❈❤ù♥❣ ♠✐♥❤ ✭✐✐✮✳ ✣➦t gn = n n−1 f ◦ Tj j=0 t❤➻ gn ≥ ✈➔ gn dµ ≤ |f | dµ ❚❤❡♦ ❜ê ✤➲ ❋❛t♦✉✬s ❝â n→∞ lim gn = |f ∗ |❧➔ ✤♦ ✤÷đ❝✱ tù❝ f L1 (X, , à) ã ự ✭✐✐✐✮✳ ❱ỵ✐ n ∈ N ✈➔ k ∈ Z✱ ✤à♥❤ ♥❣❤➽❛ Dkn = x ∈ X/ k+1 k ≤ f ∗ (x) < n n ❱ỵ✐ ♠é✐ ε > ✱ t❛ ❝â Dkn ∩ B k −ε = Dkn n ❱➻ T −1Dkn = Dkn ♥➯♥ t❛ ữủ f dà ( Dkn k )à(Dkn ) n ❱➻ ε > ❜➜t ❦➻ ✱ t❛ ❝â f dµ ≥ Dkn ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✵ k µ(Dkn ) n ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❉♦ ✤â Dkn f ∗ dµ ≤ k+1 µ(Dkn ) n ≤ n1 µ(Dkn ) + ❱➻ X = k∈Z Dkn f dµ Dkn D ✭❤đ♣ t→❝❤ ♥❤❛✉✮ ♥➯♥ t❛ ❝â X f ∗ dµ ≤ n1 µ(X) + = n + X X f dµ f dà ú ợ tt n 1✱ t❛ ❝â f ∗ dµ ≤ f dµ X X ữỡ tỹ ố ợf õ (f )dà −f dµ X X ▼➔ f ∗ dµ = f∗ dµ ≥ f dµ ❱➻ ✈➟② f ∗ dµ = f dµ ❈✉è✐ ❝ị♥❣ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ f ∗ = E(f /I) rữợ t t ú ỵ r f t õ ữủ ố ợ ■✳ ❍ì♥ ♥ú❛✱ ♥➳✉ ■ ❜➜t ❦➻ ❧➔ ❚✲❜➜t ❜✐➳♥ t❤➻ f ∗ dµ f dµ = I I ❉♦ ✤â f ∗ = E(f \I) (2) ❚ø ✭✶✮ ✈➔ ✭✷✮ t❛ ✤÷đ❝ lim n→∞ n ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ n−1 f (T j x) = E(f /I) j=0 ✸✶ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ✷✳✼ ❈→❝ ❤➺ q✉↔ ❝õ❛ ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ✷✳✼✳✶ ❈→❝ ❤➺ q✉↔ ❍➺ q✉↔ ✷✳✶✺✳ ❈❤♦ (X, β, µ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✈➔ ❝❤♦ T : X → X ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❜↔♦ t♦➔♥ ✤ë ✤♦ ❊r❣♦❞✐❝✳ ❈❤♦ f ∈ L1 (X, β, µ)✳ ❚❤➻ n n−1 f (T j x) → f dà j=0 n ợ h.k.n x X ự ỵ r ❇✐r❦❤♦❢❢ ✈ỵ✐ ■ ❧➔ σ ✲✤↕✐ sè ❝õ❛ ❝→❝ t➟♣ ❚✲❜➜t ❜✐➳♥✱ t❛ ❝â ❣✐ỵ✐ ❤↕♥ lim n→∞ n n1 f (T j x) = E(f /I) j=0 tỗ t↕✐ ✈➔ E(f /I) t❤ä❛ ♠➣♥ E(f /I) ◦ T = E(f /I) h.k.n✳ ❉♦ ❚ ❧➔ ❊r❣♦❞✐❝ ♥➯♥ E(f /I) ❧➔ ❤➔♠ ❤➡♥❣ ❤✳❦✳♥✳ ●✐↔ sû ✤â ❧➔ ❤➡♥❣ số t õ E(f /I) dà = c ì µ(X) = c ▲↕✐ ❝â E(f /I) dµ = f dµ ❉♦ ✤â f dµ = c ❱➟② lim n→∞ n n−1 f (T j x) = f dµ j=0 ❍➺ q✉↔ ✷✳✶✻✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✷ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ◆➳✉ ❚ r B t ợ − h.k.n x ∈ X ✱ t➛♥ sè ♠➔ ❝→❝ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ❇ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ µ(B)✱ ♥❣❤➽❛ ❧➔ card j ∈ {0, 1, , n − 1} /T j x ∈ B = µ(B) µ − h.k.n n→∞ n lim ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t➛♥ sè ♠➔ ❝→❝ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ❇ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ 1 lim card ≤ j ≤ n − 1/T j x ∈ B = lim n→∞ n n→∞ n n−1 χB (T j (x)) j=0 ỵ r r ợ f = B t ữủ n n n1 χB (T j (x)) = lim χB dµ = µ(B) j=0 ỵ (X, , à) ổ ❣✐❛♥ ①→❝ s✉➜t ✈➔ ❝❤♦ T : X → X ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❜↔♦ t♦➔♥ ✤ë ✤♦✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ i✳ ❚ ❧➔ ❊r❣♦❞✐❝❀ ii✳ ❱ỵ✐ ∀A, B ∈ β n n−1 µ(T −j A ∩ B) → µ(A)µ(B) n → ∞ j=0 ❈❤ù♥❣ ♠✐♥❤✳ (i) ⇒ (ii) ✿ ❣✐↔ sû r➡♥❣ ❚ ❧➔ ❊r❣♦❞✐❝✳ ❱➻ χA ∈ L1 (X, β, à) ỵ r r õ r n n−1 χA ◦ T j → µ(A) n → ∞ µ − h.k.n j=0 ◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ❝❤♦ χB ✤÷đ❝ n n−1 χA ◦ T j χB → µ(A) χB n → ∞ µ − h.k.n j=0 ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✸ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❱➻ ✈➳ tr→✐ ❜à ỵ tử trở t ữủ n n1 n à(T j A ∩ B) = j=0 = (ii) ⇒ (i)✿ ✳ ❱➟② χA ◦ T j χB dµ j=0 n−1 χA n j=0 ◦ T j χB dµ → µ(A) µ(B) n → ∞ ●✐↔ sû T −1A = A ✈➔ ❧➜② ❇❂❆✳ ❚❤➻ µ(T −j A ∩ B) = µ(A) n ✣✐➲✉ ♥➔② ❝❤♦ µ(A) ❊r❣♦❞✐❝✳ ✷✳✼✳✷ n−1 n−1 µ(A) → µ(A) n → ∞ j=0 = µ(A)2 ✳ ❉♦ ✤â µ(A) = ❤♦➦❝ ✶ ✈➔ ✈➻ ✈➟② ❚ ❧➔ Ù♥❣ ❞ö♥❣ ❈→❝ sè t➛♠ t❤÷í♥❣ ❇➜t ❦➻ sè x ∈ [0, 1] ❝â t❤➸ ✈✐➳t ♥❤÷ ♠ët sè t❤➟♣ ♣❤➙♥ ∞ x = x0 x1 x2 = j=0 xj 10j+1 ð ✤â xj ∈ {0, 1, , 9}✳ ❑❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ♥➔② ❧➔ ❞✉② ♥❤➜t ♥➳✉ ✈✐➺❝ ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ❦➳t t❤ó❝ tr♦♥❣ ✈✐➺❝ ❧➦♣ ✤✐ ❧➦♣ ❧↕✐ ✈ỉ ❤↕♥ sè ✵ ❤♦➦❝ sè ✾✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✺✿ ▼ët sè x [0, 1] ữủ t tữớ ợ ❝ì sè ✶✵✮ ♥➳✉ ♥â ❝â ♠ët ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ❞✉② ♥❤➜t ✈➔ ❝❤♦ ♠é✐ ❦❂✵✱✶✱✳ ✳ ✳ ✱✾ t❤➻ t➛♥ sè ✈ỵ✐ ❝❤ú sè ❦ ①↔② r❛ tr♦♥❣ ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ❝õ❛ ♥â ❧➔ ❜➡♥❣ ✶✴✶✵✳ ❳➨t →♥❤ ①↕ T : [0, 1] → [0, 1] ✈ỵ✐ ❚✭①✮❂✶✵① ♠♦❞ ✶✳ ❉➵ ❞➔♥❣ t❤➜② r➡♥❣✱ ✈ỵ✐ ❜➜t ❦➻ ✤è✐ sè ♥➔♦ ❝❤ó♥❣ t❛ ❝â →♥❤ ①↕ ❦➨♣✱ ✈➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ µ tr➯♥ ❬✵✱✶❪ ❧➔ ✤ë ✤♦ ❊r❣♦❞✐❝ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ ❚✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✹ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ✣➙② ❧➔ sü ❝❤✉②➸♥ t✐➳♣ ❣✐ú❛ →♥❤ ①↕ ❚ ✈➔ ❦❤❛✐ tr ữ ỵ r x [0, 1] ❝â ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ∞ x= j=0 xj = x0 x1 x2 10j+1 ❑❤✐ ✤â ∞ T (x) = 10 xj 10j+1 mod xj 10j mod j=0 ∞ = x0 + ∞ = j=0 j=0 xj+1 10j+1 = x1 x2 x3 ỵ ❍➛✉ ❤➳t ❝→❝ sè t❤ü❝ tr♦♥❣ ❬✵✱✶❪ ❧➔ t➛♠ t❤÷í♥❣ tr♦♥❣ ❝ì sè ✶✵✳ ❈❤ù♥❣ ♠✐♥❤✳ Ð ✤➙② t❛ ❝♦✐ ♥❤✐➲✉ ✤✐➸♠ ♠➔ ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ❦➳t t❤ó❝ tr♦♥❣ ✈✐➺❝ ❧➦♣ ✤✐ ❧➦♣ ❧↕✐ ✈æ ❤↕♥ ❧➛♥ sè ✵ ❤♦➦❝ ✾✳ ❉♦ ✤â✱ ✤✐➸♠ ① ❝â ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ❞✉② ♥❤➜t✳ ❈è ✤à♥❤ k ∈ {0, 1, , 9}✳ ❑❤✐ ✤â ❞➵ t❤➜② r➡♥❣ xj = k ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ T j (x) ∈ [ kr , k+1 )✳ ❱➻ ✈➟② r 1 card{0 ≤ j ≤ n−1/xj = k} = n n n−1 χ[ k , k+1 ) (T j x) r r j=0 ❚❤❡♦ ỵ r r t tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❤ë✐ tö ✈➲ χ[ k , k+1 ) (x)dx = r r r ❈❤♦ X10(k) ❜✐➸✉ t❤à t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ x ∈ [0, 1]✳ ❈❤♦ ✭✶✮ ❤ë✐ tư ❦❤✐ ✤â µ(X10(k)) = ❝❤♦ ♠é✐ ❦❂✵✱✶✱✳ ✳ ✳ ✱✾✳ ❈❤♦ X10 = X10(k).✳ ❑❤✐ ✤â µ(X10) = k=0 ◆➳✉ x ∈ X10 t❤➻ t➛♥ sè ♠➔ ❝❤ú sè ❦ ①↔② r❛ tr♦♥❣ ✈✐➺❝ ❦❤❛✐ tr✐➸♥ ♥❤à ♣❤➙♥ ❝õ❛ ① ❜➡♥❣ ✶✴✶✵✱ tù❝ ❧➔ ① ❧➔ t➛♠ t❤÷í♥❣ tr♦♥❣ ❝ì sè ✶✵✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✺ ❑✸✺●✲❑❤♦❛ ❚♦→♥ (1) ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❈❤ú sè ❤➔♥❣ ✤➛✉ ❳➨t ❞➣② (2n)n∈N : ✶✱✷✱✹✱✽✱✶✻✱✸✷✱✻✹✱✶✷✽✱✷✺✻✱✺✶✷✱✶✵✷✹✱✷✵✹✽✱✳ ✳ ✳ ❈❤ú sè ✤➛✉ t✐➯♥ ❝õ❛ sè n ∈ N ❧➔ sè ✭❣✐ú❛ sè ✵ ✈➔ sè ✾✮ ①✉➜t ❤✐➺♥ ✤➛✉ t✐➯♥ ❜➯♥ tr→✐ ❝õ❛ ♥ ❦❤✐ ♥ ✤÷đ❝ ✈✐➳t tr♦♥❣ ❝ì sè ✶✵✳ ❱➼ ❞ư✱ sè ✤➛✉ t✐➯♥ ❝õ❛ ✺✶✷ ❧➔ sè ✺✳ ❳➨t ❞➣② ❝→❝ ❝❤ú sè ❤➔♥❣ ✤➛✉✿ ✶✱✷✱✹✱✽✱✶✱✸✱✻✱✶✱✷✱✺✱✶✱✷✳ ✳ ✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ s➩ t➻♠ ①❡♠ ❝❤ú sè ❤➔♥❣ ✤➛✉ ❧➔ ❝❤ú sè ✶✱✷✱✹✳ ✳ ✳ ①✉➜t ❤✐➺♥ ✈ỵ✐ t➛♥ sè ỵ r r ú t❛ s➩ ❝❤➾ r❛ r➡♥❣ ❝❤ú sè ❤➔♥❣ ✤➛✉ ❧➔ ỳ số t ợ t số ữủ ổ tự log10 (1 + ) k rữợ t ú ỵ r ỳ số 2n tỗ t số r ≥ s❛♦ ❝❤♦ k.10r ≤ 2n < (k + 1).10r ✭✈➼ ❞ö✿2.100 ≤ 250 < 3.100 ❝❤➾ r❛ r➡♥❣ ❝❤ú sè ❤➔♥❣ ✤➛✉ ❝õ❛ ✷✺✵ ❧➔ ✷✮✳ ❙✉② r❛ log10 (k.10r ) ≤ log10 2n < log10 [(k + 1).10r ] ❙✉② r❛ log10 k + r ≤ nlog10 < log10 (k + 1) + r ❉♦ ✤â nlog10 ∈ Ik = [log10 k, log10 (k + 1)] ✣➦t log102 = α✱ ❞➣② sè (nlog10 mod 1)n∈N = 0, log10 mod 1, 2log10 mod 1, 3log10 mod 1, = 0, log10 mod 1, log10 + log10 mod 1, 2log10 + log10 mod 1, ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✻ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❧➔ q✉ÿ ✤↕♦ ❝õ❛ ✵ ✈ỵ✐ ♣❤➨♣ q✉❛② ❚ t❤❡♦ α ❉♦ ✤â Card{ N ≤ n < N s❛♦ ❝❤♦ ❝❤ú sè ✤➛✉ t✐➯♥ ❝õ❛ 2n ❧➔ k} ❂ N1 Card{ ≤ n < N s❛♦ ❝❤♦(n log10 mod 1) ∈ Ik } ❂ N1 Card{ ≤ n < N s❛♦ ❝❤♦ T n (0) ∈ Ik } ❂ N1 N −1 n=0 χIk (T n (0)) ❱➻ log10 ❧➔ ✈æ t✛ ♥➯♥ ❚ ❧➔ ♣❤➨♣ q✉❛② ✈æ t õ ỵ r r ợ f = Ik t ữủ Card{ N ≤ n < N s❛♦ ❝❤♦ ❝❤ú sè ✤➛✉ t✐➯♥ ❝õ❛ 2n ❧➔ k} N −1 χIk (T n (0)) N →∞ N n=0 = lim = χIk dµ = µ(Ik ) = log10 (k + 1) − log10 k = log10 (1 + k1 ) ▲✐➯♥ ♣❤➙♥ sè ❈❤ó♥❣ t❛ ❝â t❤➸ sû ❞ư♥❣ ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➛♥ s✉➜t ♠➔ ♠ët ❝❤ú sè ①↔② r❛ tr♦♥❣ ✈✐➺❝ ❦❤❛✐ tr✐➸♥ ❧✐➯♥ ♣❤➙♥ sè ❝õ❛ sè t❤ü❝ ✣à♥❤ ❧➼ ✷✳✶✾✳ ❍➛✉ ♥❤÷ ♠é✐ x ∈ [0, 1]✱t➛♥ sè ♠➔ ❦ sè tü ♥❤✐➯♥ ①↔② r❛ tr♦♥❣ ✈✐➺❝ ❦❤❛✐ tr✐➸♥ ❧✐➯♥ ♣❤➙♥ sè ❝õ❛ ① ❧➔ (k + 1)2 log ( ) log k(k + 2) ❈❤ù♥❣ ♠✐♥❤✿ ❈❤♦ λ ❜✐➸✉ t❤à ✤ë ✤♦ ▲❡s❜❡❣✉❡ ✈➔ µ ❜✐➸✉ t❤à ✤ë ✤♦ ●❛✉ss✱ ❦❤✐ ✤â λ✲❤✳❦✳♥ ✈➔ µ✲❤✳❦✳♥ x ∈ [0, 1] ❧➔ ✈ỉ t➾ ✈➔ ❝â ✈æ ❤↕♥ ❦❤❛✐ tr✐➸♥ ❧✐➯♥ ♣❤➙♥ sè x= ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ x0 + x1 + x ✸✼ + ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❈❤♦ ❚ ❜✐➸✉ t❤à →♥❤ ①↕ ❧✐➯♥ ♣❤➙♥ sè✳ ❑❤✐ ✤â T (x) = ❉♦ ✤â x1 + x2 + x + 1 = x1 + T (x) x2 + x3 + ]✱ ð ✤â ❬①❪ ❜✐➸✉ t❤à ♣❤➛♥ ♥❣✉②➯♥ ❝õ❛ ① ❉♦ ✤â x1 = [ T (x) ❉♦ ✤â xn = [ T x ] ❈è ✤à♥❤ k ∈ N✱ ữ ỵ r õ tr số ❜➢t ✤➛✉ ✈ỵ✐ ❝❤ú sè ❦ ✭tù❝ ❧➔ x0 = k✮ ✤ó♥❣ ❦❤✐ [ x1 ] = k ✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔✱ x0 = k ✤ó♥❣ ❦❤✐ k ≤ x1 < k + ⇔ k+1 < x ≤ k1 1 ❚ù❝ ❧➔ x ∈ ( k+1 , k1 ]✳ ❚÷ì♥❣ tü xn = k ✤ó♥❣ ❦❤✐ T n x ∈ ( k+1 , k1 ] ❉♦ ✤â n card{0 n ≤ j ≤ n − 1/xj = k} = → = = χ( log log n n−1 j=0 ,1] k+1 k χ( ,1] k+1 k (T j x) dµ [ log(1 + k1 ) − log(1 + k+1 )] (k+1) log k(k+2) ợ ✷✳✽✳✶ ▼ët sè ❜➔✐ t➟♣ ù♥❣ ❞ö♥❣ ❇➔✐ t➟♣ ✶✿ ●✐↔ sû r➡♥❣ ❚ ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❊r❣♦❞✐❝ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ①→❝ s✉➜t (X, β, µ) ✈➔ ❣✐↔ sû r➡♥❣ f ∈ L1 (X, β, µ) ✳❈✳♠✳r f (T n x) = µ − h.k.n n→∞ n lim ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✽ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ●✐↔✐✿ ●å✐ ❚ ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❊r❣♦❞✐❝ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t (X, β, µ) ✈➔ ✤➸ ❝❤♦ f ∈ L1(X, β, µ) n−1 ✣➦t Sn = f (T j x) j=0 ❚❤❡♦ r r tỗ t t s❛♦ ❝❤♦ µ(N ) = ✈➔ ♥➳✉ x ∈/ N t❤➻ Sn → n f dµ n → x / N ữ ỵ r n + Sn+1 f (T n x) Sn = + n n+1 n n ❈❤♦ n → ∞ t❛ ❝â n+1 → 1; n+1 Sn+1 → n ❉♦ ✤â ✱ ♥➳✉ x ∈/ N t❤➻ f dµ ✈➔ S n n → f dµ n → ∞ f (T n x) → n → ∞ n ❉♦ ✤â f (T n x)/n → n → ∞ cho µ − h.k.n x ∈ X ✷✳✽✳✷ ❇➔✐ t➟♣ ✷✿ ❚ø ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ❤➣② s✉② r❛ r➡♥❣ ♥➳✉ ❚ ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❊r❣♦❞✐❝ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t (X, , à) f ữủ ữ f dµ = ∞ t❤➻ n n−1 f (T j x) → ∞ µ − h.k.n j=0 ✭ ●đ✐ þ✿ ①→❝ ✤à♥❤ fM = min(f, M ) ✈➔ ❧÷✉ þ fM ∈ L1(X, β, µ)✳ ⑩♣ ❞ư♥❣ ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ❝❤♦ ♠é✐ fM ✮ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✸✾ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ●✐↔✐✿ ●✐↔ sû f ≥ ❧➔ ✤♦ ✤÷đ❝ sỷ r f dà = ố ợ ♠é✐ sè ♥❣✉②➯♥ ▼❃✵ ①→❝ ✤à♥❤ fM (x) = min{f (x), M }✱ ≤ fM ≤ M ✱ ❞♦ ✤â fM ∈ L1 (X, β, µ)✳❍ì♥ ♥ú❛ fM (x) ↑ f (x) M → ∞ ✈ỵ✐ ∀x ∈ X ❉♦ ✤â ❜➡♥❣ ✤à♥❤ ❧➼ ❤ë✐ tư ✤ì♥ ✤✐➺✉ fM dµ → f dµ = ∞ ❚❤❡♦ ✤à♥❤ ❧➼ r r õ NM X ợ à(NM ) = ữ ợ x / NM t ❝â lim n→∞ n n−1 fM (T j x) = fM dµ (1) j=0 ❈❤♦ N = ∞M =1 NM à(N ) = ỡ ỳ ố ợ ❜➜t ❦➻ ▼ ❃ ✵ t❛ ❝â✿ ♥➳✉ x ∈/ N t ú ợ K tũ ỵ ❑❤✐ fM dµ → ∞✱ ∃M > s❛♦ ❝❤♦ fM dà K õ ợ x / N t❛ ❝â lim inf n→∞ n n−1 n→∞ n n−1 f (T j x) ≥ lim j=0 fM (T j x) = fM dµ ≥ K j=0 tũ ỵ t õ ợ x ∈/ N lim inf n→∞ n ❉♦ ✤â ✈ỵ✐ µ − h.k.n n n−1 f (T j x) = ∞ j=0 ∞ f (T j x) → ∞ j=0 x ∈ X ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✹✵ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✷✳✽✳✸ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❇➔✐ t➟♣ ✸✿ ❈❤♦ r ≥ ❧➔ ♠ët sè ♥❣✉②➯♥✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❤➛✉ ♥❤÷ ♠é✐ x ∈ [0, 1]✱ xn = rn x ữủ ố ỗ ♠♦❞ ✶✳ ●✐↔✐✿ ❈❤♦ ❚✭①✮❂r① ♠♦❞ ✶✱ T : R/Z → R/Z ❚❛ ❝â ❚ ❧➔ ❊r❣♦❞✐❝ ✤è✐ ✈ỵ✐ ✤ë ✤♦ ▲❡❜❡s❣✉❡ µ✳ ❈❤♦ x ∈ [0, 1] ✈➔ ❝❤♦ xn = r n x ❱ỵ✐ {xn }✱ ♣❤➙♥ ✤♦↕♥ ♠ët ♣❤➛♥ tû ❝õ❛ xn ✳ ❈❤♦ [a, b] ⊂ [0, 1] ❈❤♦ l ∈ Z\{0} ✈➔ ✤➸ ❝❤♦ fl (x) = e2πilx ❉♦ ✤â ∃Nl ∈ β, µ(Nl ) = ✱ ❞♦ ✤â n n−1 e 2πilxn j=0 = n n−1 fl (T j x) → fl (x)dx = j=0 ✈ỵ✐ x ∈ / Nl ❈❤♦ N = l∈Z\{0} Nl ✳ ❈â µ(Nl ) = ✈➔ ✤➙② ❧➔ t➟♣ ❤đ♣ ✤➳♠ ✤÷đ❝✱ t❛ ❝â µ(N ) = ❉♦ ✤â ♥➳✉ x ∈ / N ✱ t❛ ❝â ✈ỵ✐ ∀l ∈ Z\{0} n n−1 e2πilxn = j=0 ❚❤❡♦ t✐➯✉ ❝❤✉➞♥ ❲❡②❧✱♥➳✉ x ∈ / N t❤➻ xn ✤÷đ❝ ♣❤➙♥ ❜è ỗ t x ∈ (0, 1) ❝â ❦❤❛✐ tr✐➸♥ ❧✐➯♥ ♣❤➙♥ sè x = [x0 , x1 , x2 , ] ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim n1 (x0 + x1 + + xn−1 ) = ∞ n→∞ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✹✶ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ✈ỵ✐ ❤➛✉ t x [0, 1] ủ ỵsỷ t➟♣ ✷✮ ●✐↔✐✿ ◆➳✉ x ∈ (0, 1) t❤➻ t❛ ✈✐➳t ❦❤❛✐ tr✐➸♥ ❧✐➯♥ ♣❤➙♥ sè ❝õ❛ ① ❧➔ [x0 , x1 , ] ❳➨t f : (0, 1) → R ❜ð✐ f (x) = ❑❤✐ ✤â f (x) = k ✤ó♥❣ ❦❤✐ ∞ k=1 ki χ [ ,1) k+1 k (x) < x < k1 k+1 f (x) = k ❦❤✐ x0 = k ❉♦ ✤â f (T j x) = k ✤ó♥❣ ❦❤✐ xj = k ❈❤ó♥❣ t❛ ❝â t❤➸ ✈✐➳t 1 (x0 + + xn−1 ) = n n n−1 f (T j x) j=0 ❘ã r➔♥❣ f ≥ ✈➔ ✤♦ ữủ ữ f / L1 (X, , à) ổ ss ợ f dà = log = log ≥ log ∞ k=1 ∞ k=1 ∞ kµ ( k+1 , k1 ] (k+1) k log k(k+2) k=1 k(k+1) k+2 = ∞ ❚❤❡♦ ❦➳t q✉↔ ❜➔✐ t➟♣ ✷✱ lim n→∞ n n−1 f (T j x) = ∞ j=0 ợ h.k.n x X ợ ✤♦ ●❛✉ss ✈➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❧➔ t➟♣ ❤ñ♣ ❣✐è♥❣ ♥❤÷ ✤ë ✤♦ ❦❤ỉ♥❣✳ ❚❛ ❝â lim n→∞ n n−1 f (T j (x)) = ∞ j=0 ✈ỵ✐ ❤➛✉ ❤➳t ♠å✐ ✤✐➸♠ x ∈ X ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✹✷ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ s❛✉✿ ✲✣ë ✤♦ ❊r❣♦❞✐❝✱ ♠ët sè ✈➼ ❞ö ✈➲ ✤ë ✤♦ ❊r❣♦❞✐❝✳ ✲▼ët sè ✤à♥❤ ❧➼ ❧✐➯♥ q✉❛♥✱ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ♥❤÷ ✤à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ♥â✳ ✲▼ët sè ❜➔✐ t➟♣ ù♥❣ ❞ö♥❣✳ ✣➙② ❧➔ ♥❤ú♥❣ ❝è ❣➢♥❣ ❝õ❛ ❜↔♥ t❤➙♥ ❡♠ ❞ü❛ ✈➔♦ ✈✐➺❝ t➻♠ ❤✐➸✉ ♠ët sè t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ✣➦❝ ❜✐➺t tr♦♥❣ ❦❤✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ ữủ sỹ ữợ t t ◆❣å❝ ❚r➼✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✹✸ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❍♦➔♥❣ ❚ö②✲●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐✱ ◆❤➔ ①✉➜t ❜↔♥ ❣✐→♦ ❞ö❝✳ ❬✷❪ ◆❣✉②➵♥ P❤ö ❍②✲●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ✣↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✭✶✾✾✷✮ ❬✸❪ ❇➔✐ ❣✐↔♥❣ ❝õ❛ ❉r ❈❤❛r❧❡s✳❲❛❧❦❞❡♥✲✣↕✐ ❤å❝ tê♥❣ ❤đ♣ ♠❛♥❝❤❡st❡r✳ ❳❡♠ t↕✐ ✤÷í♥❣ ❧✐♥❦✿ ✇✇✇✳♠❛t❤s✳♠❛♥❝❤❡st❡r✳❛❝✳✉❦✴ ❝✇❛❧❦❞❡♥✴❡r❣♦❞✐❝✲t❤❡♦r②✴❡r❣♦❞✐❝✲t❤❡♦r②✳❤t♠❧✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✹✹ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ... ✶✱✷✱✹✳ ✳ ✳ ①✉➜t ❤✐➺♥ ✈ỵ✐ t➛♥ số ỵ r r ❝❤ó♥❣ t❛ s➩ ❝❤➾ r❛ r➡♥❣ ❝❤ú sè ❤➔♥❣ ✤➛✉ ỳ số t ợ t số ữủ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝✿ log10 (1 + ) k rữợ t ú ỵ r ỳ số 2n tỗ t số ♥❣✉②➯♥ r ≥ s❛♦ ❝❤♦ k.10r... ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ✤➦❝ ❜✐➺t ❧➔ ỵ tt r ỵ r ự ỵ tt r ởt số q ỵ r ự õ Pữỡ ♥❣❤✐➯♥ ❝ù✉ ✣å❝ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤✱ s♦ s→♥❤✱ tê♥❣ ❤đ♣✳ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥... (T j x) j=0 ❑❤✐ ❚ ❧➔ r ố ợ t ợ ❧➔✿ n→∞ n n−1 f (T j x) = lim f dà j=0 rữợ ự t ỵ ú t ởt số t❤ù❝✿ ❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥ ✼ ❑✸✺●✲❑❤♦❛ ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥