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 ❣✐→♦ ❝æ ❣✐→♦
tr♦♥❣ tr÷í♥❣ ✣❍❙P ❍➔ ◆ë✐ ✷ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔②
tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ tî✐ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤
❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳
❉♦ ❧➛♥ ✤➛✉ t✐➯♥ ❧➔♠ q✉❡♥ ✈î✐ ❝æ♥❣ t→❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱ ❤ì♥ ♥ú❛
❞♦ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳✱ ♠➦❝ ❞ò r➜t ❝è ❣➢♥❣
♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥
✤÷ñ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥
❝õ❛ ❡♠ ✤÷ñ❝ ❤♦➔♥ 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❛ ❧✉➟♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ ❜ð✐ sü ❝è ❣➢♥❣ ♥é ❧ü❝
❝õ❛ ❜↔♥ t❤➙♥ ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✱ ❜➯♥ ❝↕♥❤ ✤â ❡♠
♥❤➟♥ ✤÷ñ❝ sü ❤÷î♥❣ ❞➝♥ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❚✳❙ ❚↕ ◆❣å❝ ❚r➼ ❝ô♥❣ ♥❤÷
❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛ t♦→♥✳
❊♠ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝
❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷ñ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳

❍➔ ◆ë✐✱ ♥❣➔② ✶✵ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✸
❙✐♥❤ ✈✐➯♥
❍♦➔♥❣ ❚❤à ▲✐➯♥

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✷

❑✸✺●✲❑❤♦❛ ❚♦→♥


▼ö❝ ❧ö❝
✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð
✶✳✶ ●✐î✐ 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 ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

✷✳✹ ❙ü tç♥ t↕✐ ❝õ❛ ❝→❝ ✤ë ✤♦ ❊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 ▲✐➯♥

✹

✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳

✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳
✳
✳

✶✽
✷✶
✷✶
✷✷
✷✹
✷✹
✷✹
✷✻
✸✷
✸✷
✸✹
✸✽
✸✽
✸✾
✹✶
✹✶

❑✸✺●✲❑❤♦❛ ❚♦→♥


õ tốt

ồ r
é

ỵ ồ t
t ởt t ồ ữủ ỹ t
ữủ ởt t ồ ờ r q tr
t tr t t ụ ữủ ởt số ở t sự
ú ỳ t q ỹ tờ qt ừ t
tt t ồ õ q sỷ ử ổ ử
t ổ tỡ õ rở ổ
ự t ồ
ợ ố ữủ ự t s s ở ổ
t s t tt r ởt số
q t r ự ử ừ õ ú ồ
ró ỡ ú ợ tr ũ ợ sỹ ừ t
sỹ ữợ t t ừ ồ r ồ t
ỵ r ởt số q tr ỵ tt r

trú ừ õ

õ ỗ ữỡ
ữỡ ổt số tự ỡ s
ữỡ r r

ử ự

ữợ q ợ ổ ự ồ t s
ỡ t t ỵ tt r ỵ r
ự ỵ tt r ởt số q ỵ
r ự ử ừ õ

Pữỡ ự

ồ t t s s tờ ủ








❈❤÷ì♥❣ ✶
▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❒
❙Ð
✶✳✶

●✐î✐ t❤✐➺✉

✳
❈❤♦ ❳ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t♦→♥ ❤å❝✳ ❳➨t →♥❤ ①↕ T : X → X ✳ ▲➜②
x ∈ X ✈➔ ❧➦♣ ❧↕✐ ù♥❣ ❞ö♥❣ ❝õ❛ →♥❤ ①↕ ❚ ✤è✐ ✈î✐ ① t❛ ✤÷ñ❝ ♠ët ❞➣②
{x, T (x), T 2 (x), T 3 (x), ...}✳ ✣➙② ❣å✐ ❧➔ q✉ÿ ✤↕♦ ❝õ❛ ①✳
◆➳✉ T n(x) = x t❤➻ ✤✐➸♠ ① ✤÷ñ❝ ❣å✐ ❧➔ t✉➛♥ ❤♦➔♥ ✈î✐ ❝❤✉ ❦➻ ♥✳
❚❛ 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♦♥❣ ❬❛✱❜❪
❧➔ ❣➻❄ ❚r÷î❝ ❤➳t t❛ ✤➣ ❜✐➳t ❤➔♠ ✤➦❝ tr÷♥❣ χA ❝õ❛ t➟♣ ❆ ✤÷ñ❝ ①→❝ ✤à♥❤
❜ð✐✿

χA =

 1
 0

✻

,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♦♥❣ ❬❛✱❜❪ ❧➔
1
n

n−1

χ[a,b] (T j (x)).
j=0

❉♦ ✤â t➛♥ sè ♠➔ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ❬❛✱❜❪ ❧➔
1
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♥❣ ❤ñ♣ ❝õ❛ ✤ë ✤♦ ❊r❣♦❞✐❝✮✿
1
lim
n→∞ n

n−1

χ[a,b] (T j (x)) = b − a
j=0

✈î✐ x ∈ X ❤✳❦✳♥✳
▼ët ❝→❝❤ tê♥❣ q✉→t ❝õ❛ ✤à♥❤ ❧➼ ♥➔② ❧➔✿ ♥➳✉ ①➨t ✈î✐ ❤➔♠ ✤♦ ✤÷ñ❝ f ❜➜t
❦➻ t❤➻ t➛♥ sè ♠➔ q✉ÿ ✤↕♦ ❝õ❛ ① ♥➡♠ tr♦♥❣ ♠ët t➟♣ ❝♦♥ A ∈ X ❧➔✿
1
lim
n→∞ n

n−1

f (T j x).
j=0

❑❤✐ ❚ ❧➔ ❊r❣♦❞✐❝ ✤è✐ ✈î✐ ✤ë ✤♦ µ t❤➻ ❣✐î✐ ❤↕♥ ♥➔② ❧➔✿
1
n→∞ n

n−1

f (T j x) =

lim

f dµ.

j=0

❚r÷î❝ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝ö t❤➸ ✤à♥❤ ❧þ ♥➔②✱ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥
t❤ù❝✿

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✼

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦
✶✳✷✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛

✣à♥❤ ♥❣❤➽❛ ✶✳✶✿ ▼ët ❧î♣ M ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳ ✤÷ñ❝ ❣å✐ ❧➔ ✤↕✐ sè
♥➳✉✿
✐✳ ∅ ∈ M;
✐✐✳ ◆➳✉ A, B ∈ M t❤➻ A ∪ B ∈ M;
✐✐✐✳ ◆➳✉ A ∈ M t❤➻ Ac ∈ M.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✿ ▼ët ❧î♣ β ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳ ✤÷ñ❝ ❣å✐ ❧➔ σ✲✤↕✐ sè ♥➳✉✿
✐✳ ∅ ∈ β;
✐✐✳ ◆➳✉ E ∈ β t❤➻ ♣❤➛♥ ❜ò ❝õ❛ ♥â X\E ∈ β;
✐✐✐✳ ◆➳✉ En ∈ β ✱ ♥❂✶✱✷✱✸✳ ✳ ✳ ❧➔ ❞➣② ✤➳♠ ✤÷ñ❝ ❝→❝ t➟♣ ❤ñ♣ tr♦♥❣ β t❤➻
∞

En ∈ β.
n=1

✣à♥❤ ♥❣❤➽❛ ✶✳✸✿ ❈❤♦ ❳ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ❝♦♠♣❛❝t✳ ▼ët t➟♣
❤ñ♣ σ ✲✤↕✐ sè ❇♦r❡❧ β(X) ✤÷ñ❝ 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

+

∪ {∞} ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ✤ë ✤♦

♥➳✉✿
✐✳ µ(∅) = 0;
✐✐✳ ◆➳✉ En ❧➔ ❝→❝ t➟♣ ❤ñ♣ ✤➳♠ ✤÷ñ❝✱ ✤æ✐ ♠ët ♣❤➙♥ ❜✐➺t tr♦♥❣ β t❤➻✿
∞

µ(

∞

En ) =
n=1

µ(En ).
n=1

❚❛ ❣å✐ (X, β, µ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✳
◆➳✉ µ(X) < ∞ t❤➻ µ ❧➔ ✤ë ✤♦ ❤ú✉ ❤↕♥✳

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✽

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

◆➳✉ µ(X) = 1 t❤➻ µ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t ✈➔ (X, β, µ) t÷ì♥❣ ù♥❣ ❧➔ ❦❤æ♥❣
❣✐❛♥ ①→❝ s✉➜t✳
✣à♥❤ ♥❣❤➽❛ ✶✳✺✿ ▼ët ❞➣② ❝→❝ ✤ë ✤♦ ①→❝ s✉➜t µn ❤ë✐ tö ②➳✉ ✤➳♥ µ ❦❤✐
n → ∞ ♥➳✉ ✈î✐ ♠é✐ f ∈ C(X, R)
f dµn →
X

f dµ
X

❦❤✐ n → ∞.

✣à♥❤ ♥❣❤➽❛ ✶✳✻✿ ❚❛ ♥â✐ ♠ët t➼♥❤ ❝❤➜t ✤ó♥❣ ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ ❳ ♥➳✉
t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ♠➔ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t ✤â ❝â ✤ë ✤♦ ✵✳

✶✳✷✳✷ ❈→❝ ✈➼ ❞ö ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦

✣ë ✤♦ ▲❡❜❡s❣✉❡ tr➯♥ ❬✵✱✶❪✳ ▲➜② ❳❂❬✵✱✶❪ ✈➔ ❧➜② M ❧➔ ❧î♣ ❝õ❛ ❝→❝

❤ñ♣ ❤ú✉ ❤↕♥ t➜t ❝↔ ❝→❝ ❦❤♦↔♥❣ ❝♦♥ ❝õ❛ ❬✵✱✶❪✳ ❱î✐ ♠é✐ ✤♦↕♥ ❝♦♥ ❬❛✱❜❪✱
✤à♥❤ ♥❣❤➽❛✿
µ ([a, b]) = b − a

❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡✳

✣ë ✤♦ ▲❡❜❡s❣✉❡ tr➯♥ R/Z✳ ▲➜② ❳❂R/Z❂❬✵✱✶✮ ♠♦❞ ✶ ✈➔ ❧➜② M ❧➔

❧î♣ ❝õ❛ ❝→❝ ❤ñ♣ ❤ú✉ ❤↕♥ t➜t ❝↔ ❝→❝ ❦❤♦↔♥❣ ❝♦♥ ❝õ❛ ❬✵✱✶✮✳ ❱î✐ ♠ët ✤♦↕♥
❝♦♥ ❬❛✱❜❪✱ ✤à♥❤ ♥❣❤➽❛✿

µ ([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✐✿

 1
δx (A) =
 0

,x ∈ A
,x ∈
/A

❚❤➻ δx ❧➔ ✤ë ✤♦ ①→❝ s✉➜t✳ ◆â ✤÷ñ❝ ❣å✐ ❧➔ ✤ë ✤♦ ❉✐r❛❝ t↕✐ ①✳

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✾

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
✶✳✸

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

❚➼❝❤ ♣❤➙♥

❈❤♦ (X, β, µ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✳

✶✳✸✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛

✣à♥❤ ♥❣❤➽❛ ✶✳✼✿ ▼ët ❤➔♠ sè f : X → R ❧➔ ✤♦ ✤÷ñ❝ ♥➳✉
f −1 [(c, ∞)] ∈ β ✈î✐ ∀c ∈ R.

✣à♥❤ ♥❣❤➽❛ ✶✳✽✿ ▼ët ❤➔♠ sè f : X → R ❧➔ ✤ì♥ ❣✐↔♥ tr➯♥ ❳ ♥➳✉ ♥â ❝â

t❤➸ ✈✐➳t ♥❤÷ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝→❝ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ t➟♣ tr♦♥❣ β ✱
♥❣❤➽❛ ❧➔

r

f=

ai χAi
i=1

✈î✐ ai ∈ R, Ai ∈ β, Ai ✤æ✐ ♠ët ❦❤æ♥❣ ❣✐❛♦ ♥❤❛✉ ✈➔ X =

r

Ai ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✾✿ ❱î✐ ♠ët ❤➔♠ ✤ì♥ ❣✐↔♥ f : X → R ✱ t❛ ❣å✐ t➼❝❤ ♣❤➙♥
❝õ❛ ❤➔♠ f tr➯♥ ❳ ❦➼ ❤✐➺✉

X

i=1

f dµ ①→❝ ✤à♥❤ ❜ð✐
r

f dµ =
X

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✿

ai µ(Ai ).
i=1

◆➳✉ f ≥ 0 t❤➻ tç♥ t↕✐ ♠ët ❞➣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 − ✈î✐ f + = max{f, 0} ≥ 0 ✈➔
f − = max{−f, 0} ≥ 0 t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ f ❜➜t ❦➻ tr➯♥

❳ ❦➼ ❤✐➺✉

X

f dµ ①→❝ ✤à♥❤ ❜ð✐
f dµ =

f + dµ −

f − dµ .

X

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✵

❑✸✺●✲❑❤♦❛ ❚♦→♥


õ tốt
ồ r
f ữủ ồ t tr
f dà < +.
X

ổ Lp


ữủ f, g : X C tữỡ ữỡ f = g h.k.n

t L1(X, , à) t ủ ợ tữỡ ữỡ ừ t
tr (X, , à) õ L1 ổ ợ

f

t d(f, g) =

f g



1

1

|f | dà.

=

t f tr tr L1(X, , à).

ợ p 1 t t ổ Lp(X, , à) ự
ữủ f : X C s |f |p t tr tr Lp(X, , à)
d(f, g) = f g p õ
f =(

1

|f |p dà) /p .

(X, , à) ổ ở ỳ 1 p < q t
Lq (X, , à) Lp (X, , à).

ở tử

ở tử ỡ
sỷ fn : X R ởt t t tr (X, , à).
fn dà ừ số tỹ t n
lim fn tỗ t
t
fn dà.

lim fn dà = lim

X n



n



X




❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
✣à♥❤ ❧➼ ✶✳✷✿✭✣à♥❤ ❧➼ ❤ë✐ tö trë✐✮

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

●✐↔ sû r➡♥❣ g : X → R ❧➔ ❦❤↔ t➼❝❤ ✈➔ fn : X → R ❧➔ ♠ët ❞➣② ❝→❝
❤➔♠ ✤♦ ✤÷ñ❝ ✈î✐ |fn | ≤ g ❤✳❦✳♥ ✈➔ lim fn = f ❤✳❦✳♥ ✳ ❚❤➻ f ❦❤↔ t➼❝❤ ✈➔
n→∞

f dµ.

fn dµ =

lim

n→∞

X

X

✶✳✹ ✣ë ✤♦ ❜➜t ❜✐➳♥
❈❤♦ (X, β, µ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳ ▼ët ♣❤➨♣ ❜✐➳♥ ✤ê✐
T : X → X ✤÷ñ❝ ❣å✐ ❧➔ ✤♦ ✤÷ñ❝ ♥➳✉ T −1 B ∈ β ✈î✐ ∀B ∈ β.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✺✿ ❚❛ ♥â✐ r➡♥❣ ❚ ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❜↔♦ t♦➔♥ ✤ë ✤♦
❤❛② µ ✤÷ñ❝ ❣å✐ ❧➔ ✤ë ✤♦ ❚✲❜➜t ❜✐➳♥ ♥➳✉ µ(T −1 B) = µ(B) ✈î✐ ∀B ∈ β.

❈❤ó þ✿ L (X, β, µ) = { f : X → R /f ✤♦ ✤÷ñ❝ ✈➔ |f | dµ < ∞}.
◆❤➟♥ ①➨t✿ µ ✤÷ñ❝ ❣å✐ ❧➔ ❚✲❜➜t ❜✐➳♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ T ∗ µ = µ✳ ❱✐➳t
1

M (X, T ) = {µ ∈ M (X)/T ∗ µ = µ}.

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✷

❑✸✺●✲❑❤♦❛ ❚♦→♥


ữỡ


ừ r
(X, , à) ởt ổ st

ởt ờ t ở õ r ởt
ờ r à ởt ở r ợ B õ
T 1 B = B à(B) = 0
ú ỵ T 1A = A ợ 0 < à(A) < 1 t õ t t T : X X
t T : A A T : (X\A) (X\A) ợ ở st t
1
1
tữỡ ự à(A)
à(. A) 1à(A)
à(. (X\A)). ổ rt
t ủ s T 1B = B à(T 1BB) = 0
õ t số ố ự
T :XX

A B = (A\B) (B\A).

B

T 1 B = B

tọ

à(T 1 BB) = 0

t tỗ t

B

à(BB) = 0 t à(B) = à (B)


ợ


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠é✐ ❥≥✵✳ ❚❛ ❝â ❜❛♦ ❤➔♠ t❤ù❝
T −j B ∆ B ⊂

j−1
i=0

j−1

=

T −(i+1) (B ∆ T −i B)
T −i (T −1 B ∆ B).

i=0

❱➻ ❚ ❜↔♦ t♦➔♥ ✤ë ✤♦ µ✱ t❛ ❝â
µ T −j B∆B ≤ jµ T −1 B∆B = 0.

✣➦t

∞ ∞

T −i B.

B∞ =
j=0 i=j

❚❛ ❝â

∞

∞
−i

i=j

❱➻

∞

T −i B

i=j

µ(B∆T −i B) = 0.

T )≤

µ(B∆

i=j

❣✐↔♠ ❦❤✐ ❥ t➠♥❣✱ ❞♦ ✤â t❛ ❝â µ(B∆ B∞) = 0✳ ◆❣♦➔✐ r❛
∞

∞ ∞

∞

T −i B = B∞ .

T −(i+1) B =

T −1 B∞ =

j=0 i=j

j=0 i=j

❍➺ q✉↔ ✷✳✷✳

◆➳✉ ❚ ❧➔ ❊r❣♦❞✐❝ ✈➔ µ(B∆ B∞) = 0 t❤➻ µ(B) = 0 ❤♦➦❝ ✶✳

✷✳✷ ✣➦❝ tr÷♥❣ ❝õ❛ ❊r❣♦❞✐❝
▼➺♥❤ ✤➲ ✷✳✸✿ ❈❤♦ ❚ ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❜↔♦ t♦➔♥ ✤ë ✤♦ ❝õ❛

(X, β, µ)✳

❈→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
i✳❚ ❧➔ ❊r❣♦❞✐❝❀
ii✳ ❇➜t ❦➻ f ∈ L1 (X, β, µ) t❤ä❛ ♠➣♥ f ◦ T = f µ✲❤✳❦✳♥ t❛ ❝â f ❧➔ ❤➔♠
❤➡♥❣ µ✲❤✳❦✳♥✳
❈❤ù♥❣ ♠✐♥❤✳

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✹

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

(i) ⇒ (ii)✳ ●✐↔ sû r➡♥❣ ❚ ❧➔ ❊r❣♦❞✐❝ ✈➔ f ∈ L1 (X, β, µ) ✈î✐ f ◦ T = f

µ ✲❤✳❦✳♥✳ ❱î✐ k ∈ Z, n ∈ N✱ ✤à♥❤ ♥❣❤➽❛
X(k, n) = {x ∈ X/

k
k+1
k k+1
≤ f (x) ≤
} = f −1 ([ n , n ]).
n
n
2
2
2
2

❱➻ f ✤♦ ✤÷ñ❝✱ X(k, n) ∈ β ✳ ❚❛ ❝â
T−1 X(k, n)∆X(k, n) ⊂ {x ∈ X/f (T x) = f (x)}.

◆➯♥
µ(T−1 X(k, n)∆X(k, n)) = 0.

❉♦ ✤â µ(X(k, n)) = 0 ❤♦➦❝ µ(X(k, n)) = 1✳
❱î✐ ♠é✐ ♥ ❝è ✤à♥❤ ✱ t❛ ❝â
µ(X∆

X(k, n)) = 0
k∈Z

✈➔ ❤ñ♣ ♥➔② ❧➔ rí✐ ♥❤❛✉✳ ❉♦ ✤â✱ t❛ ❝â
µ(X(k, n)) = µ(X) = 1

k∈Z

✈➔ ✈➻ ✈➟② ❝â ❞✉② ♥❤➜t kn ♠➔ µ(X(kn , n)) = 1✳ ❈❤♦
∞

Y=

X(kn , n).
n=1

❚❤➻ µ(Y ) = 1 ✈➔ t❤❡♦ ❝→❝❤ ①➙② ❞ü♥❣ t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣ tr➯♥ ❨✱ ♥❣❤➽❛ ❧➔✱
f ❧➔ ❤➔♠ ❤➡♥❣ µ✲❤✳❦✳♥✳
(ii) ⇒ (i) ●✐↔ sû B ∈ β ✈î✐ T −1 B = B t❤➻ t❛ ❝â χB ∈ L1 (X, β, µ) ✈➔
χB ◦ T (x) = χB (x) ∀x ∈ X ✳ ▼➔ χB ❧➔ ❤➔♠ ❤➡♥❣ µ✲❤✳❦✳♥ ✈➔ χB ❝❤➾

❧➜② ❣✐→ trà ✵ ✈➔ ✶ ♥➯♥ χB = 0µ − h.k.n ❤♦➦❝ χB = 1 h.k.n ❱➻ ✈➟②
χB dµ = 0 ❤♦➦❝ ✶✳

µ(B) =
B

❱➟② ❚ ❧➔ ❊r❣♦❞✐❝✳

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✺

❑✸✺●✲❑❤♦❛ ❚♦→♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
✷✳✸

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

❈→❝ ✈➼ ❞ö

✷✳✸✳✶ ❈→❝ ♣❤➨♣ q✉❛② ♠ët ✤÷í♥❣ trá♥
❈è ✤à♥❤ α ∈ R ✈➔ ✤à♥❤ ♥❣❤➽❛ T : R/Z → R/Z ❜ð✐
T (x) = x + α mod 1. ❚❛ ✤➣ ❜✐➳t r➡♥❣ ❚ ❜↔♦ t♦➔♥ ✤ë ✤♦ ▲❡❜❡s❣✉❡✳

✣à♥❤ ❧➼ ✷✳✹

❈❤♦ T (x) = x + α mod 1 :
i✳ ◆➳✉ α ∈ Q t❤➻ ❚ ❦❤æ♥❣ ❧➔ ❊r❣♦❞✐❝✳
ii✳ ◆➳✉ α ∈
/ Q t❤➻ ❚ ❧➔ ❊r❣♦❞✐❝✳

❈❤ù♥❣ ♠✐♥❤

✭i✮ ●✐↔ sû α

∈Q

✈➔ ✈✐➳t α = pq ✈î✐ p, q ∈ Z ✈➔ q = 0✳ ✣à♥❤ ♥❣❤➽❛
f (x) = e2πiqx ∈ L2 (X, β, µ).

●✐↔ sû ❚ ❧➔ ❊r❣♦❞✐❝✳ ❑❤✐ ✤â t❛ ❝â
f (T x) = e2πiq(x+p/q) = e2πi(qx+p) = e2πiqx = f (x).

▼➔ f ❦❤æ♥❣ ❧➔ ❤➔♠ ❤➡♥❣✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ t➼♥❤ ❝❤➜t ✷✳✸✳
❱➟② ❚ ❦❤æ♥❣ ❧➔ ❊r❣♦❞✐❝✳
✭ii✮ ●✐↔ sû r➡♥❣ α ∈/ Q✳ ●✐↔ sû f ∈ L2(X, β, µ) s❛♦ ❝❤♦ f ◦ T
❤✳❦✳♥✳ ●✐↔ sû f ❝â ❝❤✉é✐ ❋♦✉r✐❡r

=f

∞

cn e2πinx .
n=−∞

❚❤➻ f ◦ T ❝â ❝❤✉é✐ ❋♦✉r✐❡r
∞

cn e2πinα e2πinx .
n=−∞

❙♦ s→♥❤ ❝→❝ ❤➺ sè ❋♦✉r✐❡r t❛ t❤➜② r➡♥❣
cn = cn e2πinα .

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✻

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

✈î✐ ∀n ∈ Z✳ ❑❤✐ α ∈
/ Q, e2πinα = 1 trø ❦❤✐ ♥❂✵✳ ❉♦ ✤â cn = 0 ✈î✐ n = 0✳
❉♦ ✤â f ❝â ❝❤✉é✐ ❋♦✉r✐❡r c0 ✱ ♥❣❤➽❛ ❧➔✱ f ❧➔ ❤➔♠ ❤➡♥❣ ❤✳❦✳♥✳
❱➟② ❚ ❧➔ ❊r❣♦❞✐❝✳

✷✳✸✳✷

⑩♥❤ ①↕ ❦➨♣

❈❤♦ ❳❂R/Z ✈➔ ✤à♥❤ ♥❣❤➽❛ T : X → X ❜ð✐ ❚✭①✮❂✷① ♠♦❞ ✶✳
❚➼♥❤ ❝❤➜t ✷✳✺✳

⑩♥❤ ①↕ ❦➨♣ ❚ ❧➔ ❊r❣♦❞✐❝ ✤è✐ ✈î✐ ✤ë ✤♦ ▲❡❜❡s❣✉❡ µ.
❈❤ù♥❣ ♠✐♥❤✳

❈❤♦ f ∈ L2 (X, β, µ) ✈➔ ❣✐↔ sû r➡♥❣ f ◦ T = f µ✲❤✳❦✳♥✳

●✐↔ sû f ❝â ❝❤✉é✐ ❋♦✉r✐❡r
∞

am e2πimx (trong L2 ).

f (x) =
m=−∞

❱î✐ ♠é✐ j ≥ 0✱ f ◦ T j ❝â ❝❤✉é✐ ❋♦✉r✐❡r
∞
j

am e2πim2 x .
m=−∞

❙♦ s→♥❤ ❝→❝ ❤➺ sè ❋♦✉r✐❡r t❛ t❤➜②
am = a2j m

✈î✐ ∀m ∈ Z✱ ❥❂✵✱✶✱✷✳ ✳ ✳ ❜ê ✤➲ ❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡ ♥â✐ r➡♥❣ an → 0 ❦❤✐
|n| → ∞✳ ❉♦ ✤â✱ ♥➳✉ m = 0✱ t❛ ❝â am = a2j m → 0 ❦❤✐ j → ∞✳ ❉♦ ✤â ✈î✐
m = 0✱ t❛ ❝â am = 0✳ ❚ù❝ f ❝â ❝❤✉é✐ ❋♦✉r✐❡r ❧➔ a0 ✈➔ ♣❤↔✐ ❧➔ ♠ët ❤➡♥❣

sè ❤✳❦✳♥✳
❱➟② ❚ ❧➔ ❊r❣♦❞✐❝✳

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✼

❑✸✺●✲❑❤♦❛ ❚♦→♥


õ tốt


ồ r

số

T : [0, 1) [0, 1) ữủ

0 , x=0
T (x) =
1 = 1 mod 1 , 0 < x < 1
x
x

t r t ở ss
à(B) =

1
ln 2

1
dx
1+x
B

t õ t t s
t t ss t r ố ợ ở
ss

ỹ tỗ t ừ ở r
à M (X, T ) ữủ ồ ỹ tr õ
à = à1 + (1 )à2

ợ à1, à2 M (X, T ), 0 < < 1 t t õ à = à1 = à2.
s tữỡ ữỡ
ở st t à r
à ởt ỹ tr ừ
ự

i ii
ii i sỷ à ổ r
õ tỗ t B s T 1B = B 0 < à(B) < 1
ở st à1 à2 tr
à1 (A) =



à(A B)
à(A (X\B))
, à2 (A) =
.
à(B)
à(X\B)






❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

❘ã r➔♥❣ µ1 = µ2 ❞♦ µ1(B) = 1 tr♦♥❣ ❦❤✐ µ2(B) = 0✳
❱➻ T−1B = B ♥➯♥ T−1(X\B) = (X\B)✳ ❉♦ ✤â t❛ ❝â
µ1 (T −1 A) =

µ(T −1 A ∩ B)
µ(T −1 A ∩ T −1 B)

µ(A ∩ B)
=
=
= µ1 (A).
µ(B)
µ(B)
µ(B)

❚÷ì♥❣ tü
µ2 (T −1 A) =

µ(T −1 A ∩ (X\B))
= µ2 (A)
µ(X\B)

♥❣❤➽❛ ❧➔ µ1 ✈➔ µ2 ❝ò♥❣ t❤✉ë❝ ▼✭❳✱❚✮✳
❚✉② ♥❤✐➯♥✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ✈✐➳t µ ♥❤÷ tê ❤ñ♣ ❧ç✐ ❦❤æ♥❣ t➛♠ t❤÷í♥❣
µ = µ(B)µ1 + (1 − µ(B))µ2 .

❱➻ ✈➟② µ ❦❤æ♥❣ ❧➔ ❝ü❝ trà✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ ❣✐↔ t❤✐➳t✳ ❱➟② µ ❧➔
❊r❣♦❞✐❝✳
✣à♥❤ ❧➼ ✷✳✽✳ ❈❤♦ T : X → X ❧➔ ♠ët →♥❤ ①↕ ❧✐➯♥ tö❝ ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥
♠❡tr✐❝ ❝♦♠♣❛❝t t❤➻ tç♥ t↕✐ ➼t ♥❤➜t ♠ët ✤ë ✤♦ ❊r❣♦❞✐❝ tr♦♥❣ ▼✭❳✱❚✮✳

❈❤ù♥❣ ♠✐♥❤✳

❚❤❡♦ ✤à♥❤ ❧➼ ✷✳✼✱ ♥â t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ▼✭❳✱❚✮ ❝â
✤✐➸♠ ❝ü❝ trà✳
❈❤å♥ ♠ët ❞➣② ❝♦♥ ❞➔② ✤➦❝ ✤➳♠ ✤÷ñ❝ {fi}∞i=0 ❝õ❛ C(X, R)✳ ❳➨t ❤➔♠
sè ✤➛✉ t✐➯♥ f0✳ ❱➻ →♥❤ ①↕

M(X, T) → R :µ →

f0 dµ

❧➔ ❧✐➯♥ tö❝ ✈➔ ▼✭❳✱❚✮ ❧➔ ❝♦♠♣❛❝t✱ tç♥ t↕✐ v ∈ M (X, T ) s❛♦ ❝❤♦
f0 dv =

sup

f0 dµ.

µ∈M (X,T )

◆➳✉ ❝❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛
M0 = {v ∈ M (X, T )/

f0 dv =

sup

f0 dµ}

µ∈M (X,T )

t❤➻ ð tr➯♥ ✤➣ ❝❤➾ r❛ r➡♥❣ M0 ❧➔ ❦❤æ♥❣ ré♥❣✳ ▼➔ M0 ✤â♥❣✱ ❞♦ ✤â ❝♦♠♣❛❝t✳

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✶✾

❑✸✺●✲❑❤♦❛ ❚♦→♥


õ tốt

ồ r

ú t t số t t f1
M1 = {v M0 /

f1 dà}.

f1 dv = sup

àM0

ữ tr M1 ởt t ổ rộ õ ừ M0
tử q ú t
Mj = {v Mj1 /

fj dv = sup

àMj1

fj dà}

õ t ữủ ởt t ỗ
M(X, T) M0 M1 ... Mj ...

ợ ộ Mj ổ rộ õ

ớ ú t s t
j=

M =

Mj .
j=0

t M ổ rộ õ t ồ ữủ à M s r
à ỹ tr

sỷ õ t t à = à1 + (1 )à2 ợ
à1 , à2 M (X, T ), 0 < < 1.

r r à1 = à2 {fi }
i=0 tr C(X, R)
t r
fj dà2 j 0.

fj dà1 =

t f0 sỷ
f0 dà =

f0 dà1 + (1 )

f0 dà2 .

r õ
f0 dà max{

f0 dà1 ,

f0 dà2 }.




❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

▼➔ µ∞ ∈ M0 ♥➯♥
f0 dµ∞ =

f0 dµ ≥ max{

sup

f0 dµ1 ,

f0 dµ2 }.

µ∈M (X,T )

❉♦ ✤â

f0 dµ1 =

f0 dµ2 =

f0 dµ∞ .

f1 dµ2 =

f1 dµ∞ .

❚÷ì♥❣ tü ✤è✐ ✈î✐ f1 t❛ ❝â
f1 dµ1 =

❚✐➳♣ tö❝ q✉② ♥↕♣✱ ❝❤ó♥❣ t❛ ❝❤➾ r❛ r➡♥❣ ✈î✐ ❜➜t ❦➻ j ≥ 0
fj dµ1 =

fj dµ2 .

❱➟② µ∞ ❧➔ ❝ü❝ trà✳

✷✳✺ P❤➨♣ tr✉② t♦→♥ ✈➔ ❊r❣♦❞✐❝ ✤ì♥ trà

✷✳✺✳✶ ✣à♥❤ ❧➼ ♣❤➨♣ tr✉② t♦→♥ ❝õ❛ P♦✐♥❝❛r❡
❈❤♦ (X, β, µ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳

✣à♥❤ ❧➼ ✷✳✾✳✭✣à♥❤ ❧➼ ♣❤➨♣ tr✉② t♦→♥ ❝õ❛ P♦✐♥❝❛r❡✮
❈❤♦ T : X → X ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❜↔♦ t♦➔♥ ✤ë ✤♦ ❝õ❛ (X, β, µ) ✈➔ ❝❤♦
A ∈ β ❝â µ(A) > 0✳ ❚❤➻ ✈î✐ x ∈ A µ✲❤✳❦✳♥✱ q✉ÿ ✤↕♦ {T n x}∞
n=0 q✉❛② ❧↕✐ ❆
✈æ ❤↕♥ ❧➛♥✳

❈❤ù♥❣ ♠✐♥❤✳ ✣➦t E = { x ∈ A/T nx ∈ A ✈î✐ ✈æ ❤↕♥ n}
t❤➻ t❛ ♣❤↔✐ ❝❤➾ r❛ r➡♥❣ µ(A\E) = 0.
◆➳✉ t❛ ✈✐➳t
F = {x ∈ A/T n x ∈ A ∀n ≥ 1}.

❚❤➻ t❛ ❝â ✤ç♥❣ ♥❤➜t t❤ù❝
∞

(T k F ∩ A).

A\E =
k=0

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✷✶

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

❉♦ ✤â t❛ ❝â ÷î❝ ❧÷ñ♥❣
∞

T k F ∩ A)

µ(A\E) =µ(

k=0
∞

≤ µ(

T −k F )

k=0
∞

µ(T −k F ).

≤
k=0

❇➙② ❣✐í t❛ ❣✐↔ sû r➡♥❣ ♥❃♠ ✈➔ T−mF ∩T −nF = ∅ ✳ ◆➳✉ ② ♥➡♠ tr♦♥❣ ❣✐❛♦
♥➔② t❤➻ Tmy ∈ F ; T n−m(T my) = T nF ∈ F ⊂ A✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥
✈î✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❋✳ ❱➻ ✈➟② T −mF ; T −nF ❧➔ rí✐ ♥❤❛✉✳ ❱➻ {T −k F }∞n=0
❧➔ ❤å rí✐ ♥❤❛✉ ♥➯♥ t❛ ❝â
∞

∞

µ(T

−k

T −k F ) ≤ µ(X) = 1.

F ) = µ(

k=0

k=0

❱➻ µ(T−k F ) = µ(F ) ∀k ≥ 0 ✭❞♦ ✤ë ✤♦ ✤÷ñ❝ ❜↔♦ t♦➔♥✮ ♥➯♥ ❝→❝ sè ❤↕♥❣
tr♦♥❣ ♣❤➨♣ ❧➜② tê♥❣ tr➯♥ ❝â ❝ò♥❣ ❣✐→ trà ❤➡♥❣ µ(F ) ✳ ❚ù❝ ♣❤↔✐ ❝â
µ(F ) = 0.

❱➟② µ(A\E) = 0✳

✷✳✺✳✷ ❊r❣♦❞✐❝ ✤ì♥ trà

✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❈❤♦ (X, β) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ✈➔ ❝❤♦ T : X → X

❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ✤♦ ✤÷ñ❝✳ ◆➳✉ ❝â ❞✉② ♥❤➜t ✤ë ✤♦ ①→❝ s✉➜t ❚✲❜➜t
❜✐➳♥ t❤➻ t❛ ♥â✐ r➡♥❣ ❚ ❧➔ ❊r❣♦❞✐❝ ✤ì♥ trà✳

✣à♥❤ ❧➼ ✷✳✶✵

❈❤♦ ❳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ❝♦♠♣❛❝t ✈➔ ❝❤♦ T : X → X ❧➔ ♠ët ♣❤➨♣
❜✐➳♥ ✤ê✐ ❧✐➯♥ tö❝✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
i✳❚ ❧➔ ❊r❣♦❞✐❝ ✤ì♥ trà❀
ii✳ ❱î✐ ♠é✐ f ∈ C(X) tç♥ t↕✐ ♠ët ❤➡♥❣ sè ❝✭f ✮ s❛♦ ❝❤♦
n−1
1
f (T j x) → c(f ) ✤➲✉ ✈î✐ x ∈ X ❦❤✐ n → ∞✳
n
j=0

❈❤ù♥❣ ♠✐♥❤
❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✷✷

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

(ii) → (i)✿

●✐↔ sû r➡♥❣ µ✱ v ❧➔ ❝→❝ ✤ë ✤♦ ①→❝ s✉➜t ❚✲❜➜t ❜✐➳♥✱ t❛ s➩
❝❤➾ r❛ r➡♥❣ µ = v ✳ ▲➜② t➼❝❤ ♣❤➙♥ ❜✐➸✉ t❤ù❝ tr♦♥❣ ✭✐✐✮ t❛ ❝â
1
n→∞ n

n−1

f ◦ T j dµ

f dµ = lim
=
=

j=0
n−1
f

lim n1
n→∞
j=0

◦ T j dµ

c(f ) dµ = c(f ).

❚÷ì♥❣ tü ♥❤÷ t❤➳✱ t❛ ❝â
f dv = c(f ).

❉♦ ✤â
f dµ =

f dv

✈➔ ✈➻ ✈➟② µ = v.
(i) → (ii)✳ ❈❤♦ M (X, T ) = {µ}✳ ◆➳✉ ✭✐✐✮ ❧➔ ✤ó♥❣ t❤➻ t❤❡♦ ✤à♥❤ ❧➼ ❤ë✐
tö ❉♦♠✐♥❛t❡❞ t❛ ❝➛♥ ♣❤↔✐ ❝â c(f ) = f dµ✳ ●✐↔ sû ✭✐✐✮ ❧➔ s❛✐ t❤➻ ❝â t❤➸
t➻♠ f ∈ C(X) ✈➔ ❝→❝ ❞➣② nk ∈ N ✈➔ xk ∈ X s❛♦ ❝❤♦
1
lim
k→∞ nk

n−1

f (T j xk ) =

f dµ.

j=0

❱î✐ ♠é✐ k ≥ 1✱ ✤à♥❤ ♥❣❤➽❛ ♠ët ✤ë ✤♦ vk ∈ M (X) ❜ð✐
1
vk =
nk

✣➸

nk −1

T j ∗ δ xk .
j=0

1
f dvk =
nk

nk −1

f (T j xk ).
j=0

❑❤✐ ✤â vk ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉ ✤➳♥ ✤ë ✤♦ v ∈ M (X, T )✳ ✣➦❝ ❜✐➺t
t❛ ❝â
f dv = lim

k→∞

f dvk =

f dµ.

❉♦ ✤â v = µ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ ❊r❣♦❞✐❝ ✤ì♥ trà✳
❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✷✸

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

✷✳✺✳✸ ❱➼ ❞ö
❈❤♦ T = R/Z, T : X → X : x → x+α mod 1✱ α ❧➔ ✈æ t✛ t❤➻ ❚ ❧➔
❊r❣♦❞✐❝ ✤ì♥ trà ✭✈➔ µ❂ ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t ❜➜t ❜✐➳♥ ✤ì♥
trà✮✳

❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ♠ ❧➔ ♠ët ✤ë ✤♦ ①→❝ s✉➜t ❜➜t ❜✐➳♥ t❛ s➩ ❝❤➾ r❛ r➡♥❣
♠ ❂µ.
❱✐➳t ek (x) = e2πikx ✳ ❚❤➻
ek (x) dm =
=

ek (T x) dm
ek (x + α) dm

= e2πikα

ek (x) dm.

❱➻ α ❧➔ ✈æ t✛✱ ♥➳✉ k = 0 t❤➻ e2πikα = 1 ♥➯♥
ek (x) dm = 0.

❈❤♦ f ∈ C(X) ❝â ❝❤✉é✐ ❋♦✉r✐❡r

(1)
∞

ak ek ♠➔ a0 =

f dµ✳ ❱î✐ n ≥ 1✱ t❛

k=−∞

❝❤♦ σn ❜✐➸✉ t❤à ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ♥ tê♥❣ r✐➯♥❣ ✤➛✉ t✐➯♥✳ ❚❤➻ σn → f
✤➲✉ ❦❤✐ n → ∞✳ ❉♦ ✤â
lim

n→∞

σn dm =

f dm.

❚✉② ♥❤✐➯♥ sû ❞ö♥❣ ✭✶✮✱ t❛ ❝â t❤➸ t➼♥❤ t♦→♥ ✤÷ñ❝
σn dm = a0 =

❉♦ ✤â t❛ ❝â

f dm =

f dµ.

f dµ ✈î✐ ♠é✐ f ∈ C(X)✳ ❱➻ ✈➟② m = µ.

✷✳✻ ✣à♥❤ ❧➼ ❊r❣♦❞✐❝ ❝õ❛ ❇✐r❦❤♦❢❢
✷✳✻✳✶ ❑➻ ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥

✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ❈❤♦ µ ❧➔ ✤ë ✤♦ tr➯♥ (X,β)✳ ▼ët ✤ë ✤♦ v ❧➔ ❧✐➯♥ tö❝
t✉②➺t ✤è✐ ✈î✐ µ ✈➔ ✈✐➳t v ≪ µ ♥➳✉ ✈î✐ B ∈ β ♠➔ µ(B) = 0 t❛ ❧✉æ♥ ❝â

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✷✹

❑✸✺●✲❑❤♦❛ ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❚✳❙ ❚↕ ◆❣å❝ ❚r➼

v(B) = 0✳

◆❤➟♥ ①➨t✿ ◆❤÷ ✈➟② v ❧➔ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ ♥➳✉ ❝→❝ t➟♣ ❝â v✲✤ë ✤♦ ✵ t❤➻
❝ô♥❣ ❝â µ✲✤ë ✤♦ ✵ ✭♥❤÷♥❣ ❝â t❤➸ ❝â ♥❤✐➲✉ ❤ì♥ ❝→❝ t➟♣ ❝â v ✲✤ë ✤♦ ✵✮✳
❱➼ ❞ö✳ ❈❤♦ f ∈ L1 (X, β, µ) ❧➔ ❦❤æ♥❣ ➙♠ ✈➔ ✤à♥❤ ♥❣❤➽❛ ✤ë ✤♦ v ❜ð✐

v(B) =

f dµ.
B

❚❤➻ v ≪ µ.

✣à♥❤ ❧➼ ✷✳✶✶✭❘❛❞♦♥✲◆✐❦♦❞②♠✮
❈❤♦ (X, β, µ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳ ❈❤♦ v ❧➔ ♠ët ✤ë ✤♦ ✤÷ñ❝ ①→❝
✤à♥❤ tr➯♥ β ✈➔ ❣✐↔ sû r➡♥❣ v ≪ µ✳ ❚❤➻ ❝â ♠ët ❤➔♠ ✤♦ ✤÷ñ❝ ❦❤æ♥❣ ➙♠

f s❛♦ ❝❤♦
v(B) =

f dµ
B

✈î✐ ∀B ∈ β.
❍ì♥ ♥ú❛✱ f ❧➔ ❞✉② ♥❤➜t t❤❡♦ ♥❣❤➽❛ ♥➳✉ ❣ ❧➔ ❤➔♠ ✤♦ ✤÷ñ❝ ❝â ❝ò♥❣
t➼♥❤ ❝❤➜t t❤➻ f ❂❣ ❤✳❦✳♥✳
❈❤♦ M ∈ β. ❧➔ ♠ët t✐➸✉ σ ✲✤↕✐ sè✳ ❈❤ó þ r➡♥❣ µ ✤à♥❤ ♥❣❤➽❛ ♠ët ✤ë ✤♦
tr➯♥ M ❜➡♥❣ ♠ët sü ❤↕♥ ❝❤➳✳ ❈❤♦ f ∈ L1 (X, β, µ)✳ ❚❤➻ ❝❤ó♥❣ t❛ ❝â t❤➸
✤à♥❤ ♥❣❤➽❛ ♠ët ✤ë ✤♦ v tr➯♥ M ❜➡♥❣ ❝→❝❤ ❧➟♣

v(A) =

f dµ.
A

❚❛ t❤➜② v ≪ µ/A✳ ❉♦ ✤â t❤❡♦ ✤à♥❤ ❧➼ ❘❛❞♦♥✲◆✐❦♦❞②♠✱ ❝â ♠ët ❤➔♠ sè

M ✕✤♦ ✤÷ñ❝ ❞✉② ♥❤➜t E(f /M ) s❛♦ ❝❤♦
v(A) =

E(f /M ) dµ.

❚❛ ❣å✐ E(f /M ) ❧➔ ❦➻ ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ f ✤è✐ ✈î✐ σ ✲✤↕✐ sè M ✳
❚❛ t❤÷í♥❣ ✤à♥❤ ♥❣❤➽❛ E(f /M ) ✈î✐ f ❦❤æ♥❣ ➙♠ ✳✣➸ ✤à♥❤ ♥❣❤➽❛ ❝❤♦
❤➔♠ f ❜➜t ❦➻ ✱ t❛ ❝❤✐❛ f t❤➔♥❤ ♣❤➛♥ ➙♠ ✈➔ ❞÷ì♥❣ f = f+ − f− ð ✤â

❙✈✿❍♦➔♥❣ ❚❤à ▲✐➯♥

✷✺

❑✸✺●✲❑❤♦❛ ❚♦→♥