▼Ö❈ ▲Ö❈
❚r❛♥❣

▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

❈❤÷ì♥❣ ✶✳ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✸ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà ✤â♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
❈❤÷ì♥❣ ✷✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ♠ët sè ✤à♥❤ ❧þ ✈➲ sü ❤ë✐ tö ❝õ❛ ❞➣② ♣❤➛♥
tû ♥❣➝✉ ♥❤✐➯♥ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✶ ❈→❝ ❞↕♥❣ ❤ë✐ tö tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ✤â♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✷ ❈→❝ ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤❛ trà✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✶



ỵ tt st tố ởt tr ỳ ữợ ự q
trồ ừ ồ õ õ ự ử tr tỹ t r tớ
st tr õ ữợ t tr t ữủ ự
ử tr ỹ ữ tố ữ õ ồ
t t tố õ st tr t út sỹ
q t ự ừ t ồ ú t õ t t ởt số
t ồ t ự ỹ ữ r r rs
st rt r t q st tr
ởt sỹ rở tỹ sỹ t q st ỡ tr
ự ỳ ỵ õ ú tổ qt ự t

✤â✱

i) ❞➣② {xn } ⊂ X
ii)

✤÷ì❝ ❣å✐ ❧➔ ❤ë✐ tö ✈➲ x ∈ X ♥➳✉ d(xn, x) → 0 ❦❤✐ n → ∞✳
❞➣② {xn} ✤÷ñ❝ ❣å✐ ❧➔ ❞➣② ❝æs✐ ♥➳✉ ✈î✐ ♠å✐ ε > 0 tç♥ t↕✐ n0 ∈ N s❛♦ ❝❤♦
d(xn , xm ) < ε ∀ n, m ≥ n0 .

❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❝æs✐ ✤➲✉
❤ë✐ tö✳
❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ (E, . ) ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝ ❤ ♥➳✉ (E, . )
❧➔ ✤➛② ✤õ ✈î✐ d ❧➔ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ . , tù❝ ❧➔
d(x, y) = x − y

∀ x, y ∈ E.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ●✐↔ sû X ❧➔ t➟♣ ❜➜t ❦ý✱ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥

✈➔ f : X−→ E ✳ ⑩♥❤ ①↕ f ✤÷ñ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ tr➯♥ t➟♣ ❝♦♥ A ❝õ❛ X ♥➳✉ tç♥
t↕✐ ❤➡♥❣ sè c s❛♦ ❝❤♦
f (x) ≤ c ∀ x ∈ A.

✣➦t

❧➔ ❤➔♠ ❜à ❝❤➦♥ }.
✈î✐ f ∈ BE (X) ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ BE (X)✳

BE (X) = {f : X−→ E | f



f (xn ) → f (x)

❦❤✐ n → ∞.

W
❑❤✐ ✤â ❝❤ó♥❣ t❛ ❦þ ❤✐➺✉ xn −→
x✳

✣à♥❤ ❧þ ✶✳✶✳✺✳ ✐✮ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ t❤➻ t➟♣ A ⊂ X ❧➔ t➟♣
✤â♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ♠å✐ ❞➣② {xn , n ≥ 1} ⊂ A ♠➔ xn → x t❤➻ x ∈ A✳

✐✐✮

▼é✐ t➟♣ A tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ❧➔ t➟♣ ❝♦♠♣❛❝t ♥➳✉ ♠å✐ ❞➣②

{xn , n ≥ 1} ⊂ A ✤➲✉ ❝❤ù❛ ♠ët t➟♣ ❝♦♥ {xnk : k ≥ 1} ❤ë✐ tö tî✐ ♠ët ✤✐➸♠
t❤✉ë❝ ❆✳

✶✳✷✳ ❑❍➷◆● ●■❆◆ ❈⑩❈ ❚❾P ❈❖◆ ❈Õ❆ ❑❍➷◆● ●■❆◆
❇❆◆❆❈❍
●✐↔ sû (X,

.

X)

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

✶✳✷✳✶✳ ❑þ ❤✐➺✉✳

k→∞

{xnk } ⊂ {xn } : xnk −→ x ∈ A.

❱➻ B❝♦♠♣❛❝t ♥➯♥ tç♥ t↕✐
k→∞

{ynkl } ⊂ {ynk } : ynkl −→ y ∈ B.

❍✐➸♥ ♥❤✐➯♥ xn

−→ x✳

l→∞
kl

❉♦ ✤â
znkl = xnkl + ynkl → x + y

✐✐✮ A, B ❧ç✐ s✉② r❛ A + B ❧ç✐✳ ❚❤➟t ✈➟②✱ ❧➜② x, y ∈ A + B s✉② r❛
x = a1 + b1 : a1 ∈ A, b1 ∈ B

✻


✈➔
y = a2 + b2 : a2 ∈ A : b2 ∈ B.

❱➻ A ❧ç✐ ♥➯♥ ✈î✐ ♠å✐ λ ∈ [0, 1]
λa1 + (1 − λ)a2 ∈ A

✣➦❝ ❜✐➺t ❦þ ❤✐➺✉
A

K

= H(A, θ) = sup{ x

X

: x ∈ A}.

❈❤ó þ ✶✳✷✳✺✳ ◆➳✉ A, B ❧➔ ❝→❝ t➟♣ ❦❤æ♥❣ ❜à ❝❤➦♥ t❤➻ H(A, B) ❝â t❤➸ ✈æ

❤↕♥✳

✣à♥❤ ❧þ ✶✳✷✳✻✳ ❛✮ ❑❤æ♥❣ ❣✐❛♥ (Kb(X)), H) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳
❜✮ ❑❤æ♥❣ ❣✐❛♥ (Kb (X), H) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✳
❍ì♥ ♥ú❛ K(X), Kkc (X), Kbc (X) ❧➔ ❝→❝ t➟♣ ❝♦♥ ✤â♥❣ tr♦♥❣ (Kb (X), H)
❈❤ù♥❣ ♠✐♥❤✳

❛✮ ❱î✐ A, B ∈ Kb(X) t❤➻ 0 ≤ H(A, B) < ∞.
✼


❱➻ A, B ❜à ❝❤➦♥ ♥➯♥ tç♥ t↕✐ m > 0 s❛♦ ❝❤♦
A

K

≤m

❦❤✐ ✈➔ ❝❤➾ ❦❤✐


sup d(a, B) = 0

a∈A

max

sup d(a, b), sup d(b, A) = 0
a∈A

⇔


sup d(b, A) = 0

b∈B

d(a, B) = 0 ∀a ∈ A
d(b, A) = 0

∀b ∈ B

b∈B

✈➻ A, B ✤â♥❣ ♥➯♥ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ ✤ó♥❣ ❞♦ t÷ì♥❣ ✤÷ì♥❣ ✈î✐
A⊂B
B⊂A


▲➜② inf ✈î✐ c ∈ C t❛ ❝â
d(a, B) ≤ d(a, c) + inf d(c, B).
c∈C

❱➻
inf (c, B) ≤ sup d(c, B)

c∈C

c∈C

✈➔ ❧➜② sup ✈î✐ a ∈ A t❛ ❝â

sup d(a, B) ≤ sup d(a, C)+ ≤ sup d(c, B) ≤ H(A, C) + H(C, B).
a∈A

❚÷ì♥❣ tü

a∈A

c∈C

sup d(b, A) ≤ sup d(b, C)+ ≤ sup d(c, B) ≤ H(B, C) + H(C, A).
b∈B

c∈C

❙✉② r❛ H(A, B) ≤ H(A, C) + H(C, B).

✣à♥❤ ❧þ ✶✳✷✳✼✳ ◆➳✉ X ❦❤↔ ❧② t❤➻ ❦❤æ♥❣ ❣✐❛♥ (K(X, H) ❦❤↔ ❧②✳


X

+ d(xk , E).

X

X.

▲➜② inf ✈î✐ z ∈ E t❛ ❝â
◆➳✉ xk ∈ E t❤➻ d(xk , E) = 0✳
◆➳✉ xk ∈/ E ❞♦ B(xk , ε) ∩ E = ∅ t❤➻ tç♥ t↕✐ z ∈ B(xk , ε) ∩ E s❛♦ ❝❤♦
d(xk , E) = inf xk − t ≤ xk − z < ε
t∈E

s✉② r❛ d(yk , E) < 2ε ❤❛② sup d(yk , E) ≤ 2ε✳
▼➦t ❦❤→❝✱ ✈î✐ ♠é✐ y ∈ Eε ✈➔ xi ∈ N ✈î✐ i = 1, l t❛ ❝â
x − y ≤ x − xi + xi − y .

▲➜② inf ✈î✐ y ∈ Eε s✉② r❛
d(x, Eε ) ≤ x − xi + d(xi , Eε ).

▲➜② inf ✈î✐ xi ∈ Nε s✉② r❛
d(x, Eε ) ≤ d(x, Nε ) + ε.

❱➻ x ∈ E ♥➯♥ tç♥ t↕✐ i0 s❛♦ ❝❤♦ x ∈ B(xi0, ε). ❉♦ ✤â

x − x0 < ε✱

s✉② r❛

✐✐✮ ❱î✐ ♠é✐ t➟♣ ✤â♥❣ C ⊂ X, F − (C) ∈ A❀
✐✐✐✮ ❱î✐ ♠é✐ t➟♣ ♠ð O ⊂ X, F − (O) ∈ A❀
✐✈✮ ω → d(ω, F (ω)) ❧➔ ❤➔♠ ✤♦ ✤÷ñ❝ ✈î✐ ♠é✐ x ∈ X❀
✈✮ G(F ) ❧➔ A × BX ✤♦ ✤÷ñ❝✳ ❚r♦♥❣ ✤â BX ❧➔ σ ✲✤↕✐ sè ❇♦r❡❧ ❝õ❛ X✳
❑❤✐ ✤â t❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉✿
✶✮ ✭✐✮ ⇒ ✭✐✐✮ ⇒ ✭✐✐✐✮ ⇒ ✭✐✈✮ ⇒ ✭✈✮✳
✷✮ ◆➳✉ X ✤➛② ✤õ ✈➔ A ✤➛② ✤õ ✈î✐ ✤ë ✤♦ σ ✲❤ú✉ ❤↕♥ t❤➻ ✭✐✮ ⇔ ✭✐✐✮ ⇔ ✭✐✐✐✮

⇔ ✭✐✈✮ ⇔ ✭✈✮✳

✶✶


❈❍×❒◆● ✷

✣➚◆❍ ◆●❍➒❆ ❱⑨ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ❱➋ ❙Ü ❍❐■
❚Ö ❈Õ❆ ❉❶❨ P❍❺◆ ❚Û ◆●❼❯ ◆❍■➊◆ ✣❆ ❚❘➚
✷✳✶✳ ❈⑩❈ ❉❸◆● ❍❐■ ❚Ö ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❈⑩❈ ❚❾P
✣➶◆●
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ❈❤♦ {An, A} ⊂ K(X)✳
✶✳

✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö ✤➳♥ A t❤❡♦ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢✳ ❑þ ❤✐➺✉ ❧➔✿
(H)An →A ❤♦➦❝ (H) lim An = A ♥➳✉ lim H(An , A) = 0✳
n→∞
n→∞
✷✳ (An) ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö ②➳✉ ✤➳♥ A✳ ❑þ ❤✐➺✉✿ (W )An→A ❤♦➦❝
(An )

(W ) lim An = A



❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

❚❤➟t ✈➟②✱ tø d(x, A) = 0 ✈î✐ x ∈ A✳
d(x, B) = d(x, B) − d(x, A)

s✉② r❛
sup(d(x, B) − d(x, A)) = sup d(x, B)

❱➻ A ⊂ X ♥➯♥

x∈A

x∈A

sup d(x, B) ≤ sup(d(x, B) − d(x, A)).
x∈X

x∈A

✭✶✮

▼➦t ❦❤→❝✱ ❝❤♦ x ∈ X, y ∈ B ✈➔ z ∈ A t❤➻
x−y

X


x∈A

❱➟② tø ✭✶✮ ✈➔ ✭✷✮ t❛ ❝â
sup d(x, B) = sup(d(x, B) − d(x, A)).
x∈A

x∈X

❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â
sup d(x, A) = sup(d(x, A) − d(x, B)).
x∈B

x∈X

✶✸

✭✷✮


▼➔
H(A, B) = max sup d(x, B), sup d(x, A)
x∈A

x∈B

= max sup d(x, B), sup d(x, A) − d(x, B)
x∈X

x∈X



●✐↔ sû r➡♥❣ A ⊂ U (B : λ) ❦❤✐ ✤â ✈î✐ ♠å✐ ε > 0 t❤➻ tç♥ t↕✐ a0 ∈ A s❛♦ ❝❤♦
sup d(a, B) ≤ d(a0 , B) + ε.
a∈A

❚ø d(a0, B) ≤ λ t❛ ❝â sup d(a, B) ≤ λ + ε✳ ❱➻ ✈➟②
a∈A

inf{λ : A ⊂ U (B; λ)} ≥ sup d(a, B).
a∈A

❚ø ✭✶✮ ✈➔ ✭✷✮ t❛ ❝â
inf{λ : A ⊂ U (B; λ)} = sup d(a, B).
a∈A

✶✹

✭✷✮


ự tữỡ tỹ t õ
inf{ : B U (A; )} = sup d(b, A).
bB

r ự

sỷ {An, A} K(X). õ An ữủ ồ ở

tử



ó r
s lim
inf An lim sup An .
n
n

r sỹ ở tử s t ự
lim sup An A s lim
inf An .
n
n



k

k


✷✳ ❑❤→✐ ♥✐➺♠ s✲ lim
inf An ✈➔ ✇✲ lim sup An ❧➔ ❦❤→❝ ✈î✐ ❦❤→✐ ♥✐➺♠ lim inf ✈➔
n→∞
n→∞
lim sup ❧➔ ❝õ❛ ❞➣② t➟♣ {An , n ∈ N}. ▼➔ t❛ ❦þ ❤✐➺✉ ❧➔ LiAn ✈➔ LsAn ✈î✐
∞

LiAn =

An


∞

LsAn ⊂ ✇✲ lim An =
n→∞

∞

n→∞

∞

=

U
k=1 n=1 m≥n

Am ;

1
k

∞

✇✲❝❧

Am

.


(Am

pU ) .

m=n

p=1 n=1

❚r♦♥❣ ✤â ✇✲❝❧A ❧➔ ❜❛♦ ✤â♥❣ ②➳✉ ❝õ❛ t➟♣ A✳
❈❤ù♥❣ ♠✐♥❤✳

❛✮ ❚❤➟t ✈➟②
✇✲ lim sup An ⊃
n→∞

∞

✇✲ lim sup(An

pU ) .

n→∞

p=1

❚❛ ❧➜② ♠ët ❣✐→ trà x ❜➜t ❦ý x ∈ ✇✲ lim sup An t❤➻ tç♥ t↕✐ {nk } ⊂ N ✈➔ {xk }
n→∞
s❛♦ ❝❤♦ xk ∈ An (k ≥ 1) ✈➔ (W )xk →x✳
❱➟②
k

pU ) =

n→∞

n=1

∞

(Am

pU ) .

m=n

❚❤➟t ✈➟②✱ tø X∗ ❧➔ ❦❤↔ ❧②✱ ✤➦❝ ❜✐➺t ♥➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ t❤➻ pU ❧➔ ❝♦♠✲
♣❛❝t ✈➔ ♠❡tr✐❝ ❤â❛ ✤÷ñ❝ ✤è✐ ✈î✐ tæ♣æ ②➳✉✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✇✲ lim sup(An ∩
n→∞
pU ) t❛ ❝â
x ∈ ✇✲ lim sup(An
pU )
✈î✐ ♠å✐ n ≥ 1✳ ❚ø ✤â t❛ ❝â

x ∈ ✇✲❝❧

n→∞
∞

(Am

pU ) .

✷✳ ◆➳✉ dim X < ∞, {An, A} ⊂ Kk (X) ✈➔
(KM )An →A t❤➻ (H)An →A.
✶✼


❈❤ù♥❣ ♠✐♥❤✳

✶✳ ❈❤♦ x ∈ A s✉② r❛ tç♥ t↕✐ xn ∈ An ✈î✐ n ∈ N s❛♦ ❝❤♦
x − xn

X

1
< d(x, An ) + .
n

❚ø ✣à♥❤ ❧þ ✷✳✶✳✷ t❛ ❝â H(An, A) → 0 s✉② r❛ d(x, An)→d(x, A) = 0✳
❱î✐ ♠å✐ x ∈ A ❞➝♥ ✤➳♥ xn − x X→0 ❦❤✐ n→∞✳
❱➟② x ∈ s✲ lim
inf An s✉② r❛ A ⊂ s✲ lim inf An .
n→∞
n→∞
▲➜② x ∈ ✇✲ lim sup An s✉② r❛ tç♥ t↕✐ ❞➣② n1 < n2 < . . . ✈➔ xn ∈ An s❛♦
n→∞
❝❤♦ (W )xn →x✳
❚ø ❇ê ✤➲ ✷✳✶✳✼ t❛ ❝â x∗, xn ≤ S(x∗, An ) ✈î✐ ♠å✐ x∗ ∈ X∗.
▼➦t ❦❤→❝✱ tø ❣✐↔ t❤✉②➳t (H)An→A ✈➔ ❇ê ✤➲ ✷✳✶✳✼ t❛ ❝â
i

i


n→∞

❱î✐ n ∈ N ✈➔ xn ∈ An t❤➻
x − xn

X

≤ d(x, An ) +
✶✽

1
n


s✉② r❛ s✲ n→∞
lim xn → x ✈î✐ x ∈ s✲ lim inf An ✳
◆➳✉ x ∈ ✇✲ lim sup An t❤➻ tç♥ t↕✐ xk ∈ An ✈î✐ ♠å✐ k ∈ N s❛♦ ❝❤♦
n→∞
✇✲ k→∞
lim xk → x✳
❚ø ❣✐↔ t❤✐➳t dim X < +∞ t❛ ❝â s✲ k→∞
lim xk → x. ❙✉② r❛ lim d(x, An ) = 0.
k→∞
❉♦ ✤â t❛ ❝â
k

k

d(x, A) = lim d(x, An ) = lim d(x, Ank ) = 0


sup d(a, B), sup d(b, A)
a∈A
∗

b∈B
∗

= sup{|S(x , A) − S(x , B)| : x∗ ∈ S ∗ },

t❛ ❝â
sup An = sup sup S(x∗ , An ) < ∞
n∈N

n∈N x∗ ∈S ∗

✈➔ tç♥ t↕✐ K ∈ Kkc(X) s❛♦ ❝❤♦ An ⊂ K ✈î✐ ♠å✐ n ∈ N✳
✶✾


❚❛ ❝â ✈î✐ ♠å✐ n ∈ N ✈➔ x1, x2 ∈ X ∗ t❤➻
|S(x∗1 , An ) − S(x∗2 , An )| ≤ x∗1 − x∗2

X∗ .

K

K.

❙✉② r❛ i→∞


❝✮ ♥➳✉

n→∞

n→∞

✇✲lim sup An ✈➔ ✇✲lim sup Bn ❧➔ ❝→❝ t➟♣ ❦❤→❝ ré♥❣ t❤➻
n→∞

n→∞

H(✇✲ lim sup An , ✇✲ lim sup Bn ) ≤ lim sup H(An , Bn ).
n→∞

n→∞

n→∞

✶✮ ◆➳✉ x ∈ ✇✲ lim sup An t❤➻ tç♥ t↕✐ ❞➣② {xk ∈ An } s❛♦ ❝❤♦
n→∞
✇✲k→∞
lim xk = x✳ ❉♦ ✤â
❈❤ù♥❣ ♠✐♥❤✳

k

x∗ , x = lim x∗ , xk ≤ lim sup s(x∗ , An ) ≤ s(x∗ , A)
k→∞


ợ

t t õ t ồ ữủ ởt {xk : xk
x , xk lim sup s(x , An ) lim xk = x ợ x X
k
n
õ x lim sup An
x X

A nk }

s

n

lim sup s(x , An ) = x , x s(x , lim sup An ).
n

n

z lim sup (An + Bn) t tỗ t {xk An } yk Bn
n
s k
lim (xk + yk ) = z
ú t tt r k
lim xk = x lim yk = y = z x õ
x
k

k

X

lim sup xk yk
k

X

lim sup H(An , Bn ).
n

ỵ {An} Kk (X)


A=

An
n=1

ự

t n
lim H(An , A) = 0.

ợ > 0 tỗ t n N s An (A; ) ợ
(A; ) = {x x : d(x, A) < }.

ứ tt A =





sỷ



A1 (A; )
n=1

A1 t An s r tỗ t n N s A1
A(A; ) Acn ứ s r
A1 (A; )c An = .

An A1 t tt
(A; )c An =

s r An (An; ).

t õ ự

ệ ế
sỷ (, F, P) ổ st ừ E ổ
G số ừ F, B(E) số r

õ X : E tỷ

G

ữủ tr tr E X G/B(E) ữủ ợ ồ
B B(E) t X 1 (B) G). P tỷ F ữủ ữủ ồ
tỷ


❈❤ù♥❣ ♠✐♥❤✳

X, Yn −→ Y ✳
h.c.c

h.c.c
❑❤✐ ✤â Xn + Yn −→

✣➦t
Ω1 = ω : lim Xn (ω) − X(ω) = 0
n→∞

Ω2 = ω : lim Yn (ω) − Y (ω) = 0 .
n→∞

h.c.c
❚❤❡♦ ❣✐↔ t❤✐➳t Xn −→

X

h.c.c
✈➔ Yn −→

Y

♥➯♥

P(Ω1 ) = P(Ω2 ) = 1 ⇒ P(Ω1 ∩ Ω2 ) = 1.


✷✸

X +Y✳


♥➯♥
P (lim Xn + Yn − X − Y (ω) = 0) = 1.

❱➟②
h.c.c

Xn + Yn −→ X + Y.

❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
h.c.c
✣à♥❤ ❧þ ✷✳✷✳✺✳ Xn −→

X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ♠å✐ ε > 0

lim P sup Xm − X > ε = 0.

n→∞

❈❤ù♥❣ ♠✐♥❤✳

m≥n

❱î✐ ♠é✐ ε > 0 ✈➔ ♠é✐ n = 1, 2, . . . ✤➦t
∞


∀k, ∃n : Xm (ω) − X(ω) ≤ ∀m ≥ n.
k

∀k, ∃n : ω ∈

Dnc

1
k

∞

∞

k=1 n=1

❉♦ ✤â

∞

∞

Dnc

lim Xn − X = 0 =

n→∞

k=1 n=1



k=1 n=1

Dnc
n=1

Dnc
n=1

Dnc

1
k

=1

1
k

= 1 k = 1, 2, . . .

1
k

= 0 k = 1, 2, . . .

lim P(Dn ()) = 0

Dn() t õ ự



tỗ t t N p(N ) = 1 s ợ ồ N t õ
(KM )Fn ()F ().