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

ớ ỡ
ớ t tổ tọ ỏ t ỡ s s tợ ừ
t ổ tr ở ổ t t t

ụ t ũ ữớ

ữợ ự ữợ ụ ữ ở tổ õ t ỹ t õ
tổ ỷ ớ ỡ tợ t t ợ P õ
ỳ ỵ õ õ ú ù ở t t ủ tổ t õ


ỡ t

tỹ


▼ö❝ ❧ö❝
▲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❛ ❤➔♠ ✤♦ ✤÷ñ❝ ❦❤æ♥❣ ➙♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✸ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷ñ❝ ❣✐→ 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
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t ởt tr ỳ q trồ t ừ ồ
ự ử tr tỹ t ở số r t ở tỹ t ữớ t ở

t ú ợ tữủ ổ t ỹ trữợ ữủ
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t s ỡ ởt số st tr t ồ
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ự ởt số t t ừ ổ st tờ qt t

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Pữỡ ự
ữ t ồ ự t t tờ ủ tự
r ờ t ợ ữợ tr ụ ữ sr ợ tờ ở
ổ ữợ õ õ ứ õ tờ ủ tự tr
t ữỡ ự q õ tỹ t õ

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t trữớ
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ữỡ r ữỡ ú tổ tr ởt số tự
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ữỡ ữợ ự ổ st
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ữỡ r ỡ ừ t s ừ ữủ
ụ ữ t ừ ữủ tr ự tt ự ổ
st t ú tổ tr 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♥❣
tæ✐ 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) ✤÷ñ❝ ❣å✐
❧➔ σ✲ ✤↕✐ sè ❇♦r❡❧ tr♦♥❣ R✱ t❤÷í♥❣ ✤÷ñ❝ ✈✐➳t t➢t ❧➔ B(R)✳ ❈→❝ t➟♣ ♥➡♠ tr♦♥❣ B(R) ✤÷ñ❝
❣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) ✈î✐ ❜➜t ❦➻ a < b❀
✐✐✮ C2 = (−∞, a) ✈î✐ ❜➜t ❦➻ a ∈ R❀
✐✐✐✮ C3 = (a, ∞) ✈î✐ ❜➜t ❦➻ a ∈ R❀
✐✈✮ C4 = [a, b] ✈î✐ ❜➜t ❦➻ a ≤ b❀
✈✮ C5 = (−∞, a] ✈î✐ ❜➜t ❦➻ a ∈ R❀
✈✐✮ C6 = [a, ∞) ✈î✐ ❜➜t ❦➻ a ∈ R❀
✈✐✐✮ C7 = (a, b] ✈î✐ ❜➜t ❦➻ a < b❀
✈✐✐✐✮ C8 = [a, b) ✈î✐ ❜➜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 ♥➔♦ ❝ô♥❣ ❝â t❤➸ ✈✐➳t
✤÷ñ❝ ❞↕♥❣ ❤ñ♣ ✤➳♠ ✤÷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 ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ f −1(G) ∈ F ✈î✐ G ❧➔ t➟♣ ♠ð tr♦♥❣ R✳
✽


▼➺♥❤ ✤➲ ✶✳✷✳✷✳ ❍➔♠ f : X → R ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f −1(A) ∈ F ✈î✐ ♠é✐
A ∈ B(R).

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f −1(A) ∈ F

✈î✐ ♠é✐ A ∈ B(R). ❑❤✐ ✤â t❛ ❝â f −1 (G) ∈ F ✈î✐ ♠é✐

t➟♣ ♠ð G tr♦♥❣ R✱ ❞♦ ✤â f ❧➔ ❤➔♠ ❇♦r❡❧✳
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû f −1 (G) ∈ F ✈î✐ ♠é✐ t➟♣ ♠ð G tr♦♥❣ R✳ ❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ f −1 (A) ∈ F
✈î✐ ♠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 ✈î✐ 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 ✈î✐ ♠å✐ A ∈ C ✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f ❧➔ ❤➔♠ ❇♦r❡❧✳ ❑❤✐ C ⊆ B(R) t❛ ❝â f −1(A) ∈ F ✈î✐ ❜➜t ❦➻ A ∈ C ✳
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû f −1 (A) ∈ F ✈î✐ ♠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 ✈î✐ ♠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 ✈î✐
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)}

✤♦ ✤÷ñ❝✳
❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠é✐ r ∈ Q✱ t❛ ✤➦t Er = {x ∈ X : f (x) < r < g(x)}✳ ❑❤✐ ✤â
Er = {x : f (x) < r} ∩ {x : r < g(x)}

❧➔ ❣✐❛♦ ❝õ❛ ❤❛✐ t➟♣ ✤♦ ✤÷ñ❝ tr♦♥❣ X ✈➔ ✤♦ ✤â Er ✤♦ ✤÷ñ❝✳ ▼➔ E =

Er ❧➔ ❤ñ♣ ✤➳♠
r∈Q

✤÷ñ❝ ❝õ❛ ❝→❝ t➟♣ ✤♦ ✤÷ñ❝ ♥➯♥ E ✤♦ ✤÷ñ❝✳

▼➺♥❤ ✤➲ ✶✳✷✳✼✳ ❈❤♦ (X, F) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤♦ ✈➔ f : X → R, g : X → R ❧➔ ❝→❝ ❤➔♠
❇♦r❡❧✳ ❑❤✐ ✤â
✐✮ af + b ❧➔ ❤➔♠ ❇♦r❡❧ ✈î✐ ❜➜t ❦➻ a, b ∈ R❀
✐✐✮ f + g ❧➔ ❤➔♠ ❇♦r❡❧❀
✐✐✐✮ |f |α ❧➔ ❤➔♠ ❇♦r❡❧ ✈î✐ ❜➜t ❦➻ α ≥ 0❀

✐✈✮ ◆➳✉ f ❦❤æ♥❣ ❜à tr✐➺t t✐➯✉ t❤➻ f1 ❧➔ ❤➔♠ ❇♦r❡❧❀
✈✮ f g ❧➔ ❤➔♠ ❇♦r❡❧❀
✈✐✮ |f |✱max{f, g}✱min{f, g} ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧✳
❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❱î✐ ❜➜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 A ∈ F ✳ ❉♦ ✤â af + b ❧➔ ❤➔♠ ❇♦r❡❧ ✈î✐ a, b ∈ R✳
✐✐✮ ❱î✐ c ∈ R t❛ ①➨t t➟♣
B = {x : f (x) + g(x) > c} = {x : f (x) > −g(x) + c}.

❚❛ ❝â −g + c ❧➔ ❤➔♠ ❇♦r❡❧✱ ❞♦ ✤â B ∈ F ✱ tù❝ f + g ❧➔ ❤➔♠ ❇♦r❡❧✳
✐✐✐✮ ❱î✐ c ≥ 0, α ≥ 0, t❛ ①➨t t➟♣
C = {x : |f (x)|α ≤ c}
√
√
= {x : − α c ≤ f (x) ≤ α c}
√
√
= {x : − α c ≤ f (x)} ∩ {x : f (x) ≤ α c}

❚❛ ❝â C ∈ F ✳ ❱î✐ c < 0✱ t❛ ❝â C = ∅ ∈ F ✳ ❉♦ ✤â |f |α ❧➔ ❤➔♠ ❇♦r❡❧ ✈î✐ α ≥ 0✳
✐✈✮ ❱î✐ c ≥ 0✱ t❛ ①➨t t➟♣
D = {x :

❉♦ ✤â t❛ ❝â D ∈ F ❤❛②

1
≤ c} = {x : f (x) < 0} ∪ {x : 1 ≥ cf (x)} ∈ F.
f (x)
1
f

❧➔ ❤➔♠ ❇♦r❡❧✳

✈✮ ❚❛ ❝â f g = 14 (f + g)2 − 14 (f − g)2 ❧➔ ❤➔♠ ❇♦r❡❧✳
✈✐✮ t❛ ✤➣ ❝â |f |α ❧➔ ❤➔♠ ❇♦r❡❧ ✈î✐ α ≥ 0✳ ◆➳✉ ❝❤å♥ α = 1 t❤➻ t❛ ❝â |f | ❧➔ ❤➔♠ ❇♦r❡❧✳ ❚❛
①➨t
1
[f (x) + g(x)] +
2
1
min{f, g}(x) =
[f (x) + g(x)] −
2

max{f, g}(x) =

1
|f (x) − g(x)|,
2
1
|f (x) − g(x)|
2

✈î✐ ♠å✐ x ∈ X ✳
❚❛ ♥❤➟♥ t❤➜② ♠❛①{f, g} ✈➔ ♠✐♥{f, g} ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧✳

✣à♥❤ ❧þ ✶✳✷✳✽✳ ❈❤♦ (X, F) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤♦ ✈➔ {fn} ❧➔ ❞➣② ❝→❝ ❤➔♠ ❇♦r❡❧ tr➯♥ X ✳ ●✐↔
sû tç♥ t↕✐ f (x) = lim
fn (x) ✈î✐ ♠é✐ x ∈ X ✳ ❑❤✐ ✤â f ❧➔ ❤➔♠ ❇♦r❡❧✳
n

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ c ∈ R✱ t❛ s➩ ✤✐ ❝❤ù♥❣ ♠✐♥❤ t➟♣ A = {x : f (x) < c} ∈ F ✳ ❚❤➟t ✈➟②✱ ✈î✐
❜➜t ❦➻ m, n ∈ N✱ t❛ ✤➦t
Ekm = {x : fn (x) < c −
✶✶

1
, n > k}
m


❑❤✐ ✤â t❛ ❝â Ekm =
❚❛ ❝â A =
f (x) <

∞

∞

{x : fn (x) < c −
n>k

1
m}

∈ F✳

Ekm . ❚❤➟t ✈➟②✱ t❛ ❝è ✤à♥❤ x ∈ A✳ ❑❤✐ ✤â f (x) < c ✈➔ tç♥ t↕✐ m0 s❛♦ ❝❤♦

m=1 k=1
c− m10 ✳ ◆❤÷♥❣

❞♦ fn (x) → f (x) ♥➯♥ tç♥ t↕✐ k0 ∈ N s❛♦ ❝❤♦ fn (x) < c− m10 , ∀n > k0

❤❛② x ∈ Ekm00 .
❇➙② ❣✐í ❣✐↔ sû x ∈ Ekm ✈î✐ m, k ❜➜t ❦➻✳ ❑❤✐ ✤â fn (x) < c −
f (x) = lim fn (x) ≤ c −
n

1
m , ∀n

> k ✳ ✣➦❝ ❜✐➺t t❛ ❝â

1
m

✈➔ ❞♦ ✤â x ∈ A. ◆❤÷ ✈➟② t❛ ❝â
∞

∞

Ekm =

A=
m=1 k=1

Ekm .
(m,k)∈N2

A ❧➔ t➟♣ ✤➳♠ ✤÷ñ❝ ❝õ❛ ❝→❝ t➟♣ t❤✉ë❝ F ❞♦ ✤â A ∈ F ✈➔ t❛ ❝â f ❧➔ ❤➔♠ ❇♦r❡❧✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✾✳ ❈❤♦ (X, F) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤♦ ✈➔ f : X → C. ❚❛ ♥â✐ r➡♥❣ f ❧➔ ❤➔♠
❇♦r❡❧ Re❢ ✈➔ Im❢ ❧➔ ❝→❝ ❤➔♠ ❇♦r❡❧✳

▼➺♥❤ ✤➲ ✶✳✷✳✶✵✳ ❈❤♦ f : X → C✳ ❧➔ ❤➔♠ ❇♦r❡❧✳❑❤✐ ✣â |f | ❧➔ ❤➔♠ ❇♦r❡❧ ✈➔ tç♥ t↕✐ ❤➔♠
❇♦r❡❧ α : X → C ✈î✐ |α(x)| = 1, ∀x ∈ X t❤ä❛ ♠➣♥
f (x) = α(x)|f (x)|.

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠é✐ x ∈ X, f (x) = |f (x)|eiθ ✳ ❚❛ ❝â t❤➸ ❝❤å♥ θ t❤ä❛ ♠➣♥ eiθ(x) ❧➔ ❤➔♠
❇♦r❡❧✳ ❚❛ ❝â |f (x)| = (Re❢✭①✮)2 + (Im❢✭①✮)2 ❧➔ ❤➔♠ ❇♦r❡❧ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✻✳ ❚❛ ✤➦t
1
2

E = {x : f (x) = 0} = {x : |f (x)| = 0}.

❑❤✐ ✤â E ∈ F ✳ ❳➨t ❤➔♠ ❝❤➾ t✐➯✉ IE ❝õ❛ E ①→❝ ✤à♥❤ ❜ð✐

IE (x) =

1 ♥➳✉ x ∈ E
0 ♥➳✉ x ∈
/E

❧➔ ♠ët ❤➔♠ ❇♦r❡❧✳
❚❛ ✤➦t α(x) =

f (x)+IE (x)
, ∀x
|f (x)|+IE (x)

∈ X ✳ ❑❤✐ ✤â t❛ ❝â |α(x)| = 1✳ ❍ì♥ ♥ú❛ |f | + IE ❦❤æ♥❣ ❜à

tr✐➺t t✐➯✉ ✈➔ ❧➔ ❤➔♠ ❇♦r❡❧✳ ❞♦ ✤â α ❧➔ ❤➔♠ ❇♦r❡❧ ❣✐→ trà ♣❤ù❝ t❤ä❛ ♠➣♥
f (x) = α(x)|f (x)|, ∀x ∈ X.

✶✷


ởt s : X R ữủ ồ ỡ õ ỳ
ữủ ỳ tr
ữ s ỡ ợ tr 1, 2, ..., n õ
Aj = {x X : s(x) = j , j = 1, n}

t t õ

n

j IAj (x).

s(x) =
j=1

s

r ỡ Aj ữủ

ỵ f : X R r ổ õ tỗ t ởt
r ỡ ổ {sn} s
0 s1 s2 ... f

sn(x) f (x) n , x X
ự ợ n = 1, 2, ... 1 i n2n t t
En,i = {x X :

i1
i

f
(x)
<
}
2n
2n
n2n

õ Fn F En,i F, X = Fn

n = {x X : f (x) n}.

En,i t

i=1
n2n

sn (x) =
i=1

i1
IEn,i (x) + nIFn (x).
2n

õ sn r ỡ ổ tọ sn (x) f (x) ợ ộ x X
r r sn sn+1 t t sỷ r x Fn+1 õ sn+1 (x) = n + 1 > n = sn (x)
õ
sn+1 (x) =

2n+1 f (x)
j1
=
2n+1
2n+1

ợ 2n+1 f (x) ừ 2n+1 f (x) f (x) n õ 2n+1 f (x) 2n+1 n
õ t õ
sn+1 (x) =

2n+1 f (x)
2n+1 n

= n = sn (x).
2n+1
2n+1

ớ t sỷ r 0 f (x) < n õ x En,i ợ 0 i n2n sn (x) =
sỷ 2n f (x) = m + , m = 0, 1, 2, ... 0 < 1 t õ [2n f (x)] = m. õ
2n+1 f (x) = 2m + 2 ợ 0 2 < 2,


[2n f (x)]
2n .

tứ õ t ữủ
2n+1 f (x)
2m + [2]
m
[2]
=
= n + n+1
n+1
n+1
2
2
2
2
m
[2n f (x)]
n =
= sn (x).
2
2n

sn+1 (x) =

ữ t õ sn sn+1
ớ t ự sn (x) f (x) n , x X.
ợ ộ x X trữợ t t n0 ừ ợ s f (x) < n0 x
/ Fn ợ n n0
õ ợ ồ n n0 tỗ t số i õ s x En,i . õ t õ
0 f (x) sn (x) <

i
i1
1
n = n 0 n , x X.
n
2
2
2

ữ t õ sn (x) f (x) n , x X

ởt t ủ M t ừ X ởt ợ ỡ
A1 A2 ... t tr M t



Ai M.
i=1


B1 B2 ... tr M t Bi M
i=1
ừ ởt ồ t ợ ỡ ụ ợ ỡ C t tt t
ừ X t M(C) ợ ỡ s C õ ợ ỡ ọ t ự C

ỵ A ởt số t ừ X õ
M(A) = F(A).

ự t [4]

❈❤÷ì♥❣ ✷

❑❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ✈➔ ♠ët sè ♠➺♥❤ ✤➲ ✈➲ ✤ë ✤♦ ❤ú✉ ❤↕♥✱ tø ✤â ❧➔♠
❝ì sð ✤➸ ❤➻♥❤ t❤➔♥❤ ♥➯♥ t❤✉➟t ♥❣ú ✤ë ✤♦ ①→❝ s✉➜t❀ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ①→❝
s✉➜t✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ t➻♠ ❤✐➸✉ ✈➲ ✤➦❝ tr÷♥❣ ❝ì ❜↔♥ ❝õ❛ ❜✐➳♥ ❝è ♥❣➝✉ ♥❤✐➯♥ ❧➔ ❤➔♠ ♣❤➙♥
♣❤è✐ ①→❝ s✉➜t✳

✷✳✶ ✣ë ✤♦ ❤ú✉ ❤↕♥
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ▼ët ✤ë ✤♦ ❤ú✉ ❤↕♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ (X, F, µ) ❧➔ ♠ët →♥❤ ①↕
µ : F → [0, ∞)

t❤ä❛ ♠➣♥

∞

∞

An

µ

=

n=1

µ(An )
n=1

✈î✐ A1, A2, ..., An, ... ❧➔ ❞➣② ❜➜t ❦➻ ❝→❝ t➟♣ rí✐ ♥❤❛✉ tr♦♥❣ F ✳

▼➺♥❤ ✤➲ ✷✳✶✳✷✳ ❈❤♦ µ ❧➔ ✤ë ✤♦ ❤ú✉ ❤↕♥ tr➯♥ F ✳ ❑❤✐ ✤â t❛ ❝â
✐✮ µ(∅) = 0❀
✐✐✮ ◆➳✉ A1, ..., An ∈ F, ✈î✐ Ai ∩ Aj = ∅, i = j t❤➻
µ(A1 + A2 + ... + An ) = µ(A1 ) + µ(A2 ) + ... + µ(An );

✐✐✐✮ ◆➳✉ A, B ∈ F ✈➔ A ⊆ B t❤➻ µ(A) ≤ µ(B);
✐✈✮ ◆➳✉ A1 ⊆ A2 ⊆ ... ⊆ An ⊆ ... ✈î✐ An ∈ F, n = 1, 2, ... t❤➻ µ(An) ↑ µ
n → ∞❀
✈✮ ◆➳✉ A1 ⊇ A2 ⊇ ... ⊇ An ⊇ ...✱ ✈î✐ An ∈ F, n = 1, 2, ... t❤➻ µ(An) ↓ µ
n → ∞.

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ t➔✐ ❧✐➺✉ [3]✳
✶✺

Am

❦❤✐

Am

❦❤✐

m

m

▼➺♥❤ ✤➲ ✷✳✶✳✸✳ ●✐↔ sû µ : F → [0, ∞) ✈➔ µ (A ∪ B) = µ (A)+µ (B) , A, B ∈ F, A∩B =
✭tù❝ µ ❧➔ ❤ú✉ ❤↕♥ ❝ë♥❣ t➼♥❤✮✳ ❑❤✐ ✤â µ ❧➔ σ✲ ❝ë♥❣ t➼♥❤ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ µ(En) ↓ 0 ✈î✐
♠é✐ ❞➣② (En) tr♦♥❣ F t❤ä❛ ♠➣♥

∅

E1 ⊇ E2 ⊇ ...En ⊇ ...

✈➔

En = ∅.
n

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû µ ❧➔ σ✲ ❝ë♥❣ t➼♥❤✳ ❑❤✐ ✤â t❤❡♦ ♠➺♥❤ ✤➲ ✷✳✶✳✷ ✭♣❤➛♥ ✈✮ t❛ ♥❤➟♥ t❤➜②
En = ∅ t❤➻ t❛ ❝â

♥➳✉ (En ) ❧➔ ❞➣② ❣✐↔♠ tr♦♥❣ F ✈î✐
n

µ(En ) ↓ µ(∅) = 0.

◆❣÷ñ❝ ❧↕✐✱ t❛ ✤➦t (An ) ❧➔ ❞➣② ❜➜t ❦➻ tr♦♥❣ F t❤ä❛ ♠➣♥ Ai ∩ Aj = ∅, i = j ✳ ❚❛ ✤➦t
An ✈➔ En = A \ (A1 ∪ A2 ... ∪ An ) ✈î✐ n ∈ N✳ ❑❤✐ ✤â En ∈ F, En ⊇ En+1 ✈➔

A =
n

En = ∅✳ ❉♦ ✤â

n

µ(En ) ↓ µ

En

= µ(∅) = 0.

n

▼➦t ❦❤→❝ t❛ ❝â
n

µ(En ) = µ(A) − µ(A1 ∪ ... ∪ An ) = µ(A) −

µ(Ai )
i=1

✈î✐ µ ❧➔ ❤ú✉ ❤↕♥ ❝ë♥❣ t➼♥❤✳ ❚ø ✤â t❛ ❝â lim

n

n i=1

µ(Ai ) = µ(A)✱ tù❝ ❧➔ µ ❧➔ ✤➳♠ ✤÷ì❝ ❝ë♥❣

t➼♥❤✳

✷✳✷ ✣à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✈➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ µ ❧➔ ♠ët ✤ë ✤♦ ❤ú✉ ❤↕♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ (X, F, µ)✳ ◆➳✉

t❤➻ µ ✤÷ñ❝ ❣å✐ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t ✈➔ (X, F, µ) ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳
X ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➝✉✱ F ❧➔ σ ✲ ✤↕✐ sè ❝→❝ ❜✐➳♥ ❝è✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧➔ ♠ët ❤➔♠
❇♦r❡❧ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✳
µ(X) = 1

✷✳✸ ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
❳➨t ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t (Ω, S, P) ✈î✐ Ω ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➝✉✱ S ❧➔ σ ✲ ✤↕✐ sè ❝→❝ ❜✐➳♥ ❝è
✈➔ P ❧➔ ✤ë ✤♦ ①→❝ s✉➜t tr➯♥ (Ω, S), f : Ω → R ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✭ tù❝ f ❧➔ ❤➔♠ ❇♦r❡❧✮✳
❚❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉✿
✶✻


✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ f ❧➔ ❤➔♠ Ff : R → R
✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
Ff (x) = P(f ≤ x) = P({ω : f (ω) ≤ x}).
Ff

✤÷ñ❝ ①→❝ ✤à♥❤ rã ❦❤✐ {ω : f (ω) ≤ x} ∈ S, ✈î✐ x ∈ R✳

▼➺♥❤ ✤➲ ✷✳✸✳✷✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ Ff ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿
✐✮ 0 ≤ Ff (x) ≤ 1 ✈î✐ ♠å✐ x ∈ R✳
✐✐✮ Ff (x) ≤ Ff (y) ✈î✐ x ≤ y✳
✐✐✐✮ x→−∞
lim Ff (x) = 0 ✈➔ lim Ff (x) = 1.
x→+∞
✐✈✮ Ff ❧✐➯♥ tö❝ ♣❤↔✐✱ tù❝ ❧➔ ✈î✐ ♠é✐ x ∈ R t❛ ❝â Ff (x) = limh↓0 Ff (x + h)✳
❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❚❛ ❝â Ff (x) = P(f ≤ x) ∈ [0; 1] ✈î✐ ♠å✐ x✳
✐✐✮ ❱î✐ ❜➜t ❦➻ x ≤ y ✱ t❛ ❝â {ω : f (ω) ≤ x} ⊆ {ω : f (ω) ≤ y} ✈➔ ❞♦ ✤â
Ff (x) = P({ω : f (ω) ≤ x}) ≤ P({ω : f (ω) ≤ y}) = Ff (y).

✐✐✐✮ ❱î✐ n ∈ N✱ t❛ ✤➦t En = {ω : f (ω) ≤ −n}✳ ❑❤✐ ✤â E1 ⊇ E2 ⊇ ... ✈➔

En = ∅✳ ⑩P
n

❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✷ t❛ ✤÷ñ❝
Ff (−n) = P({ω : f (ω) ≤ −n}) = P(En ) → P

En

= P(∅) = 0

n

❦❤✐ n → ∞✳ ❱î✐ ε ≥ 0 ❝❤♦ tr÷î❝✱ ❝❤å♥ n0 t❤ä❛ ♠➣♥ P(En0 ) < ε✳ ❑❤✐ ✤â ✈î✐ ♠å✐ x < −n0
t❛ ❝â

0 ≤ Ff (x) ≤ Ff (−n0 ) = P(En0 ) < ε.

❉♦ ✤â lim Ff (x) = 0✳
x→−∞

✣➦t An = {ω : f (ω) ≤ n} ✈î✐ ♠å✐ n ∈ N✳ ❑❤✐ ✤â A1 ⊆ A2 ⊆ ... ✈➔

An = Ω. ❚❛ ❝â
n

Ff (n) = P({ω : f (ω) ≤ n}) = P(An ) → P

An

= P(Ω) = 1

n

❦❤✐ n → ∞✳
❱î✐ ε > 0 ❝❤♦ tr÷î❝✱ t❛ ❝❤å♥ n0 t❤ä❛ ♠➣♥ P(An0 ) > 1 − ε✳ ❑❤✐ ✤â ✈î✐ ♠å✐ x > n0 ✱ t❛ ❝â
1 ≥ Ff (x) ≥ Ff (n0 ) = P(An0 ) > 1 − ε.
✶✼


❉♦ ✤â t❛ ❝â lim Ff (x) = 1✳
x→+∞

✐✈✮ ❈è ✤à♥❤ x ∈ R✱ ✈î✐ n ∈ N t❛ ✤➦t Bn = {ω : f (ω) ≤ x+ n1 }✳ ❚❛ ❝â Bn ∈ F, B1 ⊇ B2 ⊇ ...
Bn = {ω : f (ω) ≤ x}. ▲↕✐ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✷ t❛ ✤÷ñ❝

✈➔
n

Ff (x +

1
1
) = P({ω : f (ω) ≤ x + }) = P(Bn ) → P
n
n

Bn

= Ff (x).

n

❱î✐ ε > 0 ❝❤♦ tr÷î❝✱ ❝❤å♥ n0 s❛♦ ❝❤♦

|P(Bn0 ) − Ff (x)| < ε ❤❛② |Ff (x +

❚❛ ❝â 0 ≤ Ff (x +

1
n0 ) − Ff (x)

❈❤å♥ h t❤ä❛ ♠➣♥ 0 < h <

1
n0 ✳

1
) − Ff (x)| < ε.
n0

< ε.

❑❤✐ ✤â t❛ ❝â

0 ≤ Ff (x + h) − Ff (x) ≤ Ff (x +

1
) − Ff (x) < ε.

n0

❉♦ ✤â Ff (x) = limh↓0 Ff (x + h)✳

◆❤➟♥ ①➨t ✷✳✸✳✸✳ Ff ❧➔ ♠ët ❤➔♠ t➠♥❣ tr➯♥ R ✈➔ ❜à ❝❤➦♥ ❜ð✐ 1✳ ❉♦ ✤â ♥➳✉ ❝❤♦ tr÷î❝ a ∈ R
t❛ ❝â x ↑ a t❤➻ Ff (x) t➠♥❣ ❧➯♥ ✤➳♥ ❣✐→ trà ❣✐î✐ ❤↕♥ ♥➔♦ ✤â ❝❤➥♥❣ ❤↕♥ ♥❤÷ sup Ff (x).
x◆➳✉ Ff ❧➔ ❤➔♠ t➠♥❣ t❤➻ ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ ♥â ❦❤æ♥❣ ❧î♥ ❤ì♥ Ff (a)✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ Ff (a)
❜à ❝❤➦♥ tr➯♥ t➟♣ {Ff (x) : x < a} ✈➔ ❞♦ ✤â ♥â ❧î♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ t➟♣ ♥➔②✳
◆❤÷ ✈➟② Ff ❝â ❣î✐ ❤↕♥ tr→✐ t↕✐ ♠é✐ ✤✐➸♠ t❤✉ë❝ R✱ ♥❤÷♥❣ ❣✐→ trà ❣î✐ ❤↕♥ ♥➔② ❝â t❤➸ ♥❤ä
❤ì♥ ❣✐→ trà t❤ü❝ Ff t↕✐ ✤✐➸♠ ✤â✳ ◆❣❤➽❛ ❧➔ ♥➳✉ limx↑a Ff (x) = Ff (a−) t❤➻ Ff (a−) ≤ Ff (a)✳

✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✹✳ ❇÷î❝ ♥❤↔② ❝õ❛ Ff t↕✐ a ∈ R ❧➔ ❤✐➺✉ Ff (a) − Ff (a−)✳ ❚❛ ♥â✐ r➡♥❣ a
❧➔ ♠ët ✤✐➸♠ ❧✐➯♥ tö❝ ❝õ❛ Ff ♥➳✉ Ff ❧✐➯♥ tö❝ t↕✐ a✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣ Ff (a) = F (a−) ✈➔
❞♦ ✤â ❜÷î❝ ♥❤↔② ❜➡♥❣ 0✳

▼➺♥❤ ✤➲ ✷✳✸✳✺✳ ❱î✐ ❜➜t ❦➻ a ∈ R✱ ❜÷î❝ ♥❤↔② ❝õ❛ Ff t↕✐ a ❜➡♥❣ P(f = a)✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ t❛ ❝â
Ff (a) = P(f ≤ a) = P({ω : f (ω) ≤ a}).

✣➦t An = {ω : f (ω) ≤ a − n1 } ✈î✐ n ∈ N✳ ❚❛ ❝â
A1 ⊆ A2 ⊆ ... ✈➔

An = {ω : f (ω) < a}.
n

✶✽


t õ

Ff (a

1
1
) = P({ : f () a }) = P(An ) P
n
n

An

= P(f < a).

n

ữ t õ lim Ff (a n1 ) = Ff (a) õ P(f < a) = Ff (a) ứ õ t õ ữợ
n



Ff (a) Ff (a) = P(f a) P(f < a) = P(f = a).

õ ởt t ủ ữủ õ ỳ ỗ t rộ ổ
ữủ õ t t tữỡ ự ợ t số tỹ N).

ữợ 0 ừ ố Ff t t ởt t
ữủ
ự sỷ J t ữợ 0 ừ Ff

t

Jn = {a J : ữợ ừ Ff t

1
}.
n

sỷ a1 , a2 , ..., ak Jn ợ a1 < a2 < ... < ak . ồ a0 số tỹ t s
a0 < a1 õ 0 Ff (a0 ) Ff (ak ) 1 t t t ừ Ff õ t õ
Ff (ai ) Ff (ai1 ) n1 õ
k

1 Ff (ak ) Ff (a0 ) =
i=1

1
Ff (ai ) Ff (ai1 ) k .
n

ứ õ t õ k n õ Jn t rộ t ổ ự ỡ n
tỷ t t õ J =

Jn J ữủ
n

q ợ t f tỗ t t ữủ J R s
P(f = x) = 0 ợ ồ x R \ J

ự ợ f

t J R t ữợ 0 ừ Ff

õ J ởt t ữủ õ t õ P(f = x) = 0 ợ x J õ P(f = x) = 0
ợ x R \ J.



ỵ ợ f t trữợ tọ t tr 2.3.2
ổ tỗ t ởt ở st tr (R, B(R)) tọ F (x) = ((, x]) ợ ồ
x R ỡ ỳ t
õ ởt số t q s
((, a]) = F (a),
((a, b]) = F (b) F (a+) ợ a < b,
((a, b)) = F (b) F (a+) ợ a < b,
([a, b)) = F (b) F (a) ợ a < b
([a, b]) = F (b) F (a) ợ a b.

ở st s F ữủ ồ ở s tts s F
t sỷ F ữủ
F (x) =


0,

x,
1,



x<0
0x1

x>1

t t õ
((, 0]) = F (0) = 0,
((1, )) = (R) ((, 1]) = 1 F (1) = 0.

ợ 0 a b 1, t õ
((a, b )) = ([a, b)) = (( a, b]) = ([a, b]) = b a.

ú t ởt ở

ỵ sỷ t õ F tọ t t tr 2.3.2 õ
tỗ t ởt f s F = Ff
ự rữợ t t s r ởt ổ st ỡ f

ữủ

t = R, S = B(R) P ở st s stts tr (R, B(R)) s
F

ởt f tr (R, B(R), P) f (x) = x. õ f



❇♦r❡❧ ✈➔ t❛ ❝â
Ff (a) = P(f ≤ a)
= P({x ∈ R : f (x) ≤ a})
= P({x ∈ R : x ≤ a})
= P((−∞, a])
= F (a).

◆❤÷ ✈➟② t❛ ❝â F = Ff ✳

◆❤➟♥ ①➨t ✷✳✸✳✶✵✳ ❱î✐ ❜➜t ❝ù ❤➔♠ ♣❤➙♥ ♣❤è✐ F

♥➔♦ ❝❤♦ tr÷î❝ ❧✉æ♥ tç♥ t↕✐ ♠ët ❜✐➳♥

♥❣➝✉ ♥❤✐➯♥ ♥➔♦ ✤â ♥❤➟♥ F ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ ❝❤♦ ❤➔♠
x
t2
dt
e− 2 √
2π

F (x) =
−∞

t❤➻ tç♥ t↕✐ ♠ët ❜✐➳♥ ❝è ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ F ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② t❛
♥â✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳

✷✶


❈❤÷ì♥❣ ✸

❚➼❝❤ ♣❤➙♥
❈❤÷ì♥❣ ♥➔② ♣❤→t tr✐➸♥ ❤ì♥ ♥ú❛ ❧þ t❤✉②➳t t➼❝❤ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ❤ú✉ ❤↕♥
(X, F, µ) ❜➜t ❦➻✳ ❚ø ✤â ❧➔♠ ♥➲♥ t↔♥❣ ❝❤♦ ♠ët sè ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t tr♦♥❣ ❝→❝ ❝❤÷ì♥❣

t✐➳♣ t❤❡♦✳

✸✳✶ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤ì♥ ❣✐↔♥ ❦❤æ♥❣ ➙♠
✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✶✳ ●✐↔ sû s =

n

αi IAi
i=1

❧➔ ♠ët ❤➔♠ ✤ì♥ ❣✐↔♥ ✳ ❱î✐ E ∈ F ✱ t❛ ✤à♥❤ ♥❣❤➽❛

t➼❝❤ ♣❤➙♥ ❝õ❛ s tr➯♥ E t❤❡♦ µ ❧➔

n

αi µ(Ai ∩ E).

s dµ =
i=1

E

✣➦❝ ❜✐➺t ✈î✐ A ∈ F ✱ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ❝❤➾ t✐➯✉ ❝õ❛ A tr➯♥ X ❝❤➼♥❤ ❧➔ ✤ë ✤♦ ❝õ❛ A✿
IA dµ = µ(A).
X

◆➳✉ µ ❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ tr➯♥ [0, 1] ✈➔ A = [a, b] ✈î✐ 0 ≤ a ≤ b ≤ 1 t❤➻ t❛ ❝â
I[a,b] dµ = b − a
[0,1]

t❤÷í♥❣ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ ❝õ❛ I[a,b] tr➯♥ [0, 1]✳
◆➳✉ X = R ✈➔ µ ❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ ✲ ❙t✐❡❧t❥❡s tr➯♥ R s✐♥❤ ❜ð✐ ❤➔♠
x

F (x) =

ρ(t)dt
−∞

✈î✐ ρ ❧➔ ❤➔♠ ❦❤↔ t➼❝❤ ❘✐❡♠❛♥♥ ❦❤æ♥❣ ➙♠✱ ❦❤✐ ✤â ✈î✐ a < b t❛ ❝â
b

I[a,b] dµ = µ([a, b]) = F (b) − F (a) =

ρ(t)dt.
a

R

✷✷


✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✷✳ ❈❤♦ ❤➔♠ f : X → R ✤♦ ✤÷ñ❝ ✈➔ ❣✐↔ sû f ≥ 0✳ ❱î✐ ❜➜t ❦➻ E ∈ F ✱ t❛
✤à♥❤ ♥❣❤➽❛
f dµ = sup
E

s dµ
E

✈î✐ s ❧➔ ❤➔♠ ✤ì♥ ❣✐↔♥ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ 0 ≤ s(x) ≤ f (x), ∀x ∈ X ✳ ◆➳✉ ✈➳ ♣❤↔✐ ❦❤æ♥❣
❤ú✉ ❤↕♥✱ t❛ ♥â✐ f ❦❤æ♥❣ ❦❤↔ t➼❝❤ tr➯♥ E ✳

▼➺♥❤ ✤➲ ✸✳✶✳✸✳ ●✐↔ sû f, g ❧➔ ❝→❝ ❤➔♠ ✤♦ ✤÷ñ❝ ✈➔ E ∈ F ✳ ❑❤✐ ✤â
✐✮ ◆➳✉ 0 ≤ f ≤ g t❤➻
f dµ ≤

✐✐✮ ◆➳✉ A ⊆ B, A, B ∈ F, f ≥ 0 t❤➻

E

g dµ.
E

f dµ ≤
A

f dµ.
B

✐✐✐✮ ◆➳✉ f (x) = 0 ✈î✐ ♠å✐ x ∈ E t❤➻
f dµ = 0
E

✐✈✮ ◆➳✉ f (x) ≥ 0, c ≥ 0, c ❧➔ ❤➡♥❣ sè t❤➻
cf dµ = c
E

f dµ.
E

✈✮ ◆➳✉ µ(E) = 0, f ≥ 0 t❤➻
f dµ = 0.
E

✈✐✮ ◆➳✉ f ≥ 0 t❤➻

f dµ =
E

IE f dµ.
X

❈❤ù♥❣ ♠✐♥❤✳ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✸❪✳

▼➺♥❤ ✤➲ ✸✳✶✳✹✳ ❈❤♦ s, t ❧➔ ❝→❝ ❤➔♠ ✤ì♥ ❣✐↔♥ ❜➜t ❦➻ ✈î✐ s ≥ 0, t ≥ 0✳ ❱î✐ ♠é✐ E ∈ F ✱
✤➦t ϕ(E) =

sdµ.

❑❤✐ ✤â ϕ ❧➔ ✤ë ✤♦ ❤ú✉ ❤↕♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤♦ (X, F)✳ ❍ì♥ ♥ô❛ t❛ ❝â

E

(s + t)dµ =
X

sdµ +
X

❈❤ù♥❣ ♠✐♥❤✳ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✸❪✳
✷✸

tdµ.
X


✸✳✷ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷ñ❝ ❦❤æ♥❣ ➙♠
✣à♥❤ ❧þ ✸✳✷✳✶✳ ✭✣à♥❤ ❧þ ▲❡❜❡s❣✉❡ ✈➲ sü ❤ë✐ tö ✤ì♥ ✤✐➺✉✮✳ ❈❤♦ (fn) ❧➔ ♠ët ❞➣②
❝→❝ ❤➔♠ ✤♦ ✤÷ñ❝ tr➯♥ X ✈➔ ❣✐↔ sû
✐✮ 0 ≤ f1(x) ≤ f2(x) ≤ ... ✈î✐ x ∈ X ✱
✐✐✮ fn(x) → f (x) ❦❤✐ n → ∞, ✈î✐ x ∈ X ✳
❑❤✐ ✤â f ✤♦ ✤÷ñ❝ ✈➔
fn dµ →
X

❦❤✐ n → ∞✳

f dµ
X

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ t➔✐ ❧✐➺✉ [3]✳

❍➺ q✉↔ ✸✳✷✳✷✳ ●✐↔ sû f ≥ 0 ✈➔ g ≥ 0✱ f ✈➔ g ❦❤↔ t➼❝❤✳ ❑❤✐ ✤â f + g ❦❤↔ t➼❝❤ ✈➔
(f + g)dµ =
X

f dµ +
X

gdµ.
X

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ t➔✐ ❧✐➺✉ ❬✸❪✳

✸✳✸ ❚➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷ñ❝ ❣✐→ trà ♣❤ù❝
✣à♥❤ ♥❣❤➽❛ ✸✳✸✳✶✳ ❍➔♠ ❣✐→ trà ♣❤ù❝ f tr➯♥ X ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ t➼❝❤ ✭▲❡❜❡s❣✉❡✮ t❤❡♦ µ
♥➳✉ |f | ❦❤↔ t➼❝❤✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ f ♥❤÷ t❤➳ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ L1(X, µ)✳
❱î✐ f = u + iv ∈ L1(X, µ)✱ t❛ ❝â
u+ dµ −

f dµ =
X

X

v+ dµ − i

u− dµ + i
X

X

v− dµ.
X

✣à♥❤ ❧þ ✸✳✸✳✷✳ ◆➳✉ f, g ∈ L1(X, µ) ✈➔ a, b ∈ C✱ ❦❤✐ ✤â af + bg ∈ L1(X, µ) ✈➔
(af + bg)dµ = a
X

f dµ + b
X

gdµ.
X

❈❤ù♥❣ ♠✐♥❤✳ ❉♦ |af + bg| ≤ |a||f | + |b||g| ♥➯♥ |af + bg| ❦❤↔ t➼❝❤✱ ❞♦ ✤â af + bg ∈ L1(X, µ)
✈➔ t❛ ❝â
|af + bg|dµ ≤ |a|
X

|f |dµ + |b|
X

✷✹

|g|dµ.
X


✣➛✉ t✐➯♥ t❛ ❣✐↔ sû r➡♥❣ f ✈➔ g ❧➔ ❝→❝ ❤➔♠ ❣✐→ trà t❤ü❝✳ ✣➦t h = f + g ✱ ❦❤✐ ✤â
h = h+ − h− = f+ − f− + g+ − g−

✈➔ ❞♦ ✤â
h+ + f− + g− = h− + f+ + g+ ,

tø ✤â t❛ ❝â
(h+ + f− + g− )dµ =
X

(h− + f+ + g+ )dµ.
X

⑩♣ ❞ö♥❣ ❍➺ q✉↔ ✸✳✷✳✷ t❛ ❝â
h+ dµ +
X

f− dµ +
X

g− dµ =
X

h− dµ +
X

f+ dµ +
X

g+ dµ
X

❤❛②
h+ dµ −
X

f+ dµ −

h− dµ =
X

X

g+ dµ −

f− dµ +
X

X

g− dµ
X

❙✉② r❛
hdµ =
X

f dµ +
X

gdµ.
X

❇➙② ❣✐í ❣✐↔ sû f ✈➔ g ❧➔ ❝→❝ ❤➔♠ ❣✐→ trà ♣❤ù❝ ✈î✐ f = Re❢ + ✐Im❢✱❣ = Re❣ + ✐Im❣✱ ✤➦t
h = f + g, h = Re❤ + ✐Im❤✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ ❝â
Re❤ ❞µ + ✐

hdµ =
X

X

X

Im❢ ❞µ,
X

Re❣ ❞µ + ✐

gdµ =
X

X

Re❢ ❞µ + ✐

f dµ =
X

Im❤ ❞µ,

X

Im❣ ❞µ,
X

◆❤÷♥❣ t❛ ❝â
Re❤ = Re❢ + Re❣, Im❤ = Im❢ + Im❣,

✈➔

(Re❢ + Re❣)❞µ =
X

Re❢❞µ +
X

(Im❢ + Im❣)❞µ =
X

Re❣❞µ,
X

Im❢❞µ +
X

✷✺

Im❣❞µ.
X