▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà ✤â♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ởt số ỵ ✈➲ sü ❤ë✐ tö ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶ ❈→❝ ❞↕♥❣ ❤ë✐ tư tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ✤â♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❈→❝ ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤❛ trà✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tt st tố ởt tr ỳ ữợ ự q trồ ❝õ❛ ❚♦→♥ ❤å❝✱ ♥â ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ t❤ü❝ t➳✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✱ ①→❝ s✉➜t ✤❛ trà õ ữợ t tr t ữủ ù♥❣ ❞ư♥❣ tr➯♥ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷✿ tè✐ ÷✉ ❤â❛ ✈➔ ✤✐➲✉ ❦❤✐➸♥✱ ❤➻♥❤ ❤å❝ ♥❣➝✉ ♥❤✐➯♥✱ t♦→♥ ❦✐♥❤ t➳✱ t❤è♥❣ ❦➯✱✳✳✳ ❱➻ ❧➩ ✤â✱ ①→❝ s✉➜t ✤❛ trà ✤➣ t❤✉ ❤ót sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝✳ ❈❤ó♥❣ t❛ ❝â t❤➸ ❦➸ t➯♥ ♠ët sè ♥❤➔ t♦→♥ ❤å❝ t✐➯✉ ❜✐➸✉ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❧➽♥❤ ✈ü❝ ♥➔② ♥❤÷✿ ●❡r❛❧❞ ❇❡❡r✱ ❈❤❛r❧❡s ❈❛st❛✐♥❣✱ ❋✉♠✐♦ ❍✐❛✐✱ ❘♦❜❡rt ▲❡❡ ❚❛②❧♦r✱✳✳✳ ❈→❝ ❦➳t q✉↔ ✈➲ ①→❝ s✉➜t ✤❛ trà ❧➔ ♠ët sü ♠ð rë♥❣ t❤ü❝ sü ❝→❝ ❦➳t q✉↔ ✈➲ ①→❝ s✉➜t ✤ì♥ trà✳ ❈➠♥ ❝ù ✈➔♦ ♥❤ú♥❣ ỵ õ ú tổ qt ự t➔✐ ✏❈→❝ ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ tr ữủ tỹ ữợ sỹ ữợ t t P ❱➠♥ ◗✉↔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ ❚❤➛②✳ ◆❤➙♥ ❞à♣ ♥➔② t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝ ✈➔ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣ t→❝ ❣✐↔ ①✐♥ ỡ ỗ tr ợ ❈❛♦ ❤å❝ ✶✼ ✲ ❳→❝ s✉➜t t❤è♥❣ ❦➯ ✤➣ ❝ë♥❣ t→❝ ✈➔ ❣✐ó♣ ✤ï tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ✷ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ rt ữủ ỳ ỵ õ ❣â♣ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tèt ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ❣✐↔ ✸ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ❞➣② {xn} ⊂ X✳ ❑❤✐ ✤â✱ i) ❞➣② {xn } ⊂ X ii) ✤÷ì❝ ❣å✐ ❧➔ ❤ë✐ tö ✈➲ x ∈ X ♥➳✉ d(xn, x) → ❦❤✐ n → ∞✳ ❞➣② {xn} ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❝ỉs✐ ♥➳✉ ✈ỵ✐ ♠å✐ ε > tỗ t n0 N s d(xn , xm ) < ε ∀ n, m ≥ n0 ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❝ỉs✐ ✤➲✉ ❤ë✐ tư✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ (E, ) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝ ❤ ♥➳✉ (E, ) ❧➔ ✤➛② ✤õ ✈ỵ✐ 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 ✤÷đ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ tr t A X tỗ t sè c s❛♦ ❝❤♦ f (x) ≤ c ∀ x ∈ A ✣➦t ❧➔ ❤➔♠ ❜à ❝❤➦♥ } ✈ỵ✐ f ∈ BE (X) ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ BE (X)✳ BE (X) = {f : X−→ E | f ❑❤✐ ✤â✱ f = sup f (x) x∈X ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ●✐↔ sû {fn} ⊂ BE (X)✱ X ❧➔ t➟♣ ❜➜t ❦ý✳ õ {fn} ữủ tợ ❤➔♠ f tr➯♥ X ♥➳✉ ✈ỵ✐ ♠å✐ ε > x X tỗ t n0 = n0(, x) s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ≥ n0 t❤➻ i) ❤ë✐ tử fn (x) f (x) < ỵ fn → f ❦❤✐ n → ∞ ❤❛② n→∞ lim fn (x) = f (x) ii) ❤ë✐ tö t❤❡♦ ❝❤✉➞♥ sup tr♦♥❣ BE (X) tỵ✐ f ♥➳✉ f ∈ BE (X) ✈➔ sup fn (x) − f (x) → ❦❤✐ n → ∞ x∈X ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ●✐↔ sû {xn} ❧➔ ❞➣② tr♦♥❣ ❦❤æ♥❣ ✤à♥❤ ❝❤✉➞♥ E ✳ ❉➣② ữủ tử tợ x {xn} ❤ë✐ tư ✤➳♥ x ∈ X t❤❡♦ tỉ♣ỉ ②➳✉ C(E, E ∗)✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ f ∈ E ∗ t❤➻ {xn } f (xn ) → f (x) n W õ ú t ỵ xn x ỵ sỷ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ t❤➻ t➟♣ A ⊂ X ❧➔ t➟♣ ✤â♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠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✐ tư tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ ❆✳ ✶✳✷✳ ❑❍➷◆● ●■❆◆ ❈⑩❈ ❚❾P ❈❖◆ ❈Õ❆ ❑❍➷◆● ●■❆◆ ❇❆◆❆❈❍ ●✐↔ sû (X, X) ❧➔ ổ ỵ P0(x) ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ X ✐✐✮ Kkc ❧➔ t t ỗ rộ X ✐✐✐✮ Kb(X) ❧➔ ❤å ❝→❝ t➟♣ ❝♦♥ ✤â♥❣✱ ❜à ❝❤➦♥✱ ỗ rộ X Kbc(X) t ỗ rộ X ✈✮ K(X) ❧➔ ❤å ❝→❝ t➟♣ ❝♦♥ ✤â♥❣ ❦❤→❝ ré♥❣ ❝õ❛ X✳ ✈✐✮ ❚r➯♥ P0(X) ✤à♥❤ ♥❣❤➽❛ ❤❛✐ ♣❤➨♣ t♦→♥ ữ s ợ A, B P0(X) ∈ R A + B = {a + b : a ∈ A, b ∈ B} λA = {λa : a A}, R ỵ A B = ❝❧{a + b : a ∈ A, b ∈ B} ❝❧✿ ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ A + B tr X ỵ A, B Kkc(X) t❤➻ A + B ∈ Kkc(X)✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ A, B ❝♦♠♣❛❝t t❤➻ A + B ❝♦♠♣❛❝t✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ A + B ❝♦♠♣❛❝t t❛ ❝❤ù♥❣ ♠✐♥❤ ♠å✐ ❞➣② {zn} ⊂ A + B ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö✳ ✐✮ ●✐↔ sû {zn} ⊂ A + B õ tỗ t {xn} A, {yn} B s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ zn = xn + yn At tỗ t k {xnk } {xn } : xnk −→ x ∈ A ❱➻ B❝♦♠♣❛❝t tỗ t k {ynkl } {ynk } : ynkl −→ y ∈ B ❍✐➸♥ ♥❤✐➯♥ xn −→ x✳ l→∞ kl ❉♦ ✤â znkl = xnkl + ynkl → x + y A, B ỗ s r A + B ỗ t x, y A + B s✉② r❛ x = a1 + b1 : a1 ∈ A, b1 ∈ B ✻ ✈➔ y = a2 + b2 : a2 ∈ A : b2 ∈ B A ỗ ợ [0, 1] λa1 + (1 − λ)a2 ∈ A λb1 + (1 − λ)b2 ∈ A ❑❤✐ ✤â λx + (1 − λ)y = λ(a1 + b1 ) + (1 − λ)(a2 + b2 ) ∈ A + B ❙✉② r❛ A + B ỗ A, B ∈ P0(X), x ∈ X t❤➻ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ x ✈➔ A ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ d(x, A) = inf d(x, y) y∈A ✈ỵ✐ d(x, y) = x−y X ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ ❑❤♦↔♥❣ ❝→❝❤ tr➯♥ P0(X) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ H(A, B) = max sup d(a, B), sup d(b, A) a∈A bB t ỵ A K = H(A, ) = sup{ x X : x A} ú ỵ ✶✳✷✳✺✳ ◆➳✉ A, B ❧➔ ❝→❝ t➟♣ ❦❤æ♥❣ ❜à ❝❤➦♥ t H(A, B) õ t ổ ỵ ❛✮ ❑❤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♥❣ ♠✐♥❤✳ ❛✮ ❱ỵ✐ A, B ∈ Kb(X) t❤➻ ≤ H(A, B) < A, B tỗ t↕✐ m > s❛♦ ❝❤♦ A K ≤m B K ≤ m ▲➜② a ∈ A s✉② r❛ d(a, B) = inf (a, y) ≤ inf d(a, 0) + inf d(0, y) ≤ 2m y∈B ▲➜② b ∈ B s✉② r❛ d(b, A) = inf (b, y) ≤ inf d(b, 0) + inf d(0, y) ≤ 2m y∈A ❉♦ ✤â H(A, B) = max sup d(a, B), sup d(b, A) < ∞ a∈A b∈B ❱➻ A ✤â♥❣ s✉② r❛ A = A = {x ∈ X : d(x, A) = 0}✳ ◆➳✉ A ⊂ B t❤➻ d(x, B) ≤ d(x, A) ✈ỵ✐ ♠å✐ x ∈ X ❚❛ ❝â H(A, B) = ⇔ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐  sup d(a, B) =  a∈A max sup d(a, b), sup d(b, A) = a∈A ⇔  sup d(b, A) = b∈B d(a, B) = ∀a ∈ A d(b, A) = ∀b ∈ B b∈B ✈➻ A, B ✤â♥❣ ♥➯♥ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ ✤ó♥❣ ❞♦ tữỡ ữỡ ợ AB BA s r A = B tø ✤à♥❤ ♥❣❤➽❛ s✉② r❛ H(A, B) = H(B, A) ❱ỵ✐ A, B, C ∈ Kb (X) s✉② r❛ H(A, B) ≤ H(A, C) + H(C, B) ✽ ❚❤➟t ✈➟②✱ ❧➜② a ∈ A, b ∈ B, c ∈ C t❛ ❝â a−b ≤ a−c X + c−b d(a, B) ≤ a − c X + d(c, B) X X ▲➜② inf ✈ỵ✐ b ∈ B t❛ ❝â ▲➜② inf ✈ỵ✐ 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 ✈ỵ✐ 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) ỵ X ❦❤↔ ❧② t❤➻ ❦❤æ♥❣ ❣✐❛♥ (K(X, H) ❦❤↔ ❧②✳ ▲➜② ♠ët t➟♣ ❝♦♥ D ✤➳♠ ✤÷đ❝ trị ♠➟t tr♦♥❣ X✳ ●å✐ D ❧➔ t➟♣ ❝→❝ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ D✳ ❑❤✐ ✤â D ❧➔ ✤➳♠ ✤÷đ❝✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ D ❧➔ trị ♠➟t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ K(X)✳ l ❱ỵ✐ ♠é✐ E ∈ K(X) ✈➔ ε > 0, tỗ t {xi E : 1, l} s E ⊂ B(xi, ε) i=1 ✈➔ E ∩ B(xi , ε) = ∅ ✈ỵ✐ i = 1, l, tr♦♥❣ ✤â B(xi, ε) ❧➔ ❤➻♥❤ ❝➛✉ ♠ð t➙♠ xi ❜→♥ ❦➼♥❤ D trũ t tr X tỗ t↕✐ Eε = {y1, y2, , yl } ⊂ D s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ x k − yk < ε ✾ ✈ỵ✐ k = 1, l ❘ã r➔♥❣ Eε ∈ D✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ H(Eε, E) < 2ε ❚❤➟t ✈➟②✱ ✈ỵ✐ z ∈ E t❤➻ yk − z ≤ yk − x k X + xk − z d(yk , E) ≤ yk − xk X + d(xk , E) X X ▲➜② inf ✈ỵ✐ z ∈ E t❛ ❝â ◆➳✉ xk ∈ E t❤➻ d(xk , E) = 0✳ ◆➳✉ xk ∈/ E ❞♦ B(xk , ) E = t tỗ 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 ❦❤→❝✱ ✈ỵ✐ ♠é✐ y ∈ Eε ✈➔ xi ∈ N ✈ỵ✐ i = 1, l t❛ ❝â x − y ≤ x − xi + xi − y ▲➜② inf ✈ỵ✐ y ∈ Eε s✉② r❛ d(x, Eε ) ≤ x − xi + d(xi , Eε ) ▲➜② inf ✈ỵ✐ xi ∈ Nε s✉② r❛ d(x, Eε ) ≤ d(x, N ) + x E tỗ t↕✐ i0 s❛♦ ❝❤♦ x ∈ B(xi0, ε) ❉♦ ✤â x − x0 < ε✱ s✉② r❛ d(x, Eε ) < ε ❱➻ t❤➳ sup d(x, Eε) ≤ 2c✳ ❉♦ ✈➟② H(Eε, E) < 2ε x∈X ✶✳✸✳ P❍❺◆ ❚Û ◆●❼❯ ◆❍■➊◆ ◆❍❾◆ ●■⑩ ❚❘➚ ✣➶◆● ●✐↔ sû (Ω, A) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤♦ ✤÷đ❝✱ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ▼ët →♥❤ ①↕ F : Ω → P0(X) ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤❛ trà✳ ✶✵ ▼➔ 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 = sup |d(x, A) − d(x, B)| x∈X ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ( ❈❤ó þ✿ sup |f (x)| = max sup(f (x)), sup(−f (x)) ) xX xX ú ỵ ứ ỵ tr t❛ ❝â ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ ✤÷ì♥❣ ✈➲ ❦❤♦↔♥❣ ❝→❝❤ ❍❛✉s❞♦r❢❢ H(A, B) = max{inf{λ : B ⊂ U (A; λ)}; inf{λ : A ⊂ U (B; λ)}}, ✈ỵ✐ U (A; λ) = {x : d(x, A) ≤ λ} ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ ❤✐➸♥ ♥❤✐➯♥ A ⊂ U (B : sup d(a, B)) t❤➻ t❛ ❝â a∈A inf{λ : A ⊂ U (B : λ)} ≤ sup d(a, B) a∈A ✭✶✮ ●✐↔ sû r➡♥❣ A ⊂ U (B : λ) ❦❤✐ õ ợ > t tỗ 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 ✶✹ ✭✷✮ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â inf{λ : B ⊂ U (A; λ)} = sup d(b, A) b∈B ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✹✳ ●✐↔ sû {An, A} ⊂ K(X) ❑❤✐ ✤â✱ An ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tö ✤➳♥ A t❤❡♦ ♥❣❤➽❛ ▼♦s❝♦ ♥➳✉ ✇✲ lim sup An = A = s✲ lim inf An , n→∞ n→∞ tr♦♥❣ ✤â ✇✲ lim sup An = {x = ✇✲ lim xm : xm ∈ Am, m ∈ M ✈ỵ✐ M ⊂ N} n→∞ ✈➔ s✲ lim inf An = {x = s✲ lim xn : xn An , n N} n ỵ ❧➔ (KM )An→A ❤♦➦❝ (KM ) n→∞ lim An = A✳ ❈❤ó♥❣ t❛ t❤➜② r➡♥❣ s✲ lim inf xn = x ❝â ♥❣❤➽❛ ❧➔ xn − x X = ✈➔ ✇✲ lim xm = x ❝â ♥❣❤➽❛ ❧➔ xm ❤ë✐ tö ②➳✉ ✤➳♥ x✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ i) x lim sup An tỗ t ❝♦♥ {An } ⊂ {An } ✈➔ xn ∈ An s❛♦ W ❝❤♦ xn −→ x✳ ii) x ∈ s lim inf An tỗ t {xn }; xn ∈ An s❛♦ ❝❤♦ xn −→ x✳ k k ✣à♥❤ ❧➼ ✷✳✶✳✺✳ ✶✳ ❘ã r➔♥❣ s✲ lim inf An ⊂ ✇✲ lim sup An n→∞ n→∞ ❱➻ ✈➟② ❦❤✐ ❝❤➾ r❛ sü ❤ë✐ tö ▼♦s❝♦ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✇✲ lim sup An ⊂ A ⊂ s✲ lim inf An n→∞ n→∞ ✶✺ k k ✷✳ ❑❤→✐ ♥✐➺♠ s✲ lim inf An ✈➔ ✇✲ lim sup An ❧➔ ❦❤→❝ ✈ỵ✐ ❦❤→✐ ♥✐➺♠ lim inf ✈➔ n→∞ n→∞ lim sup ❧➔ ❝õ❛ ❞➣② t➟♣ {An , n N} t ỵ LiAn LsAn ✈ỵ✐ ∞ LiAn = An ∞ ✈➔ LsAn = k=1 n≥k An k=1 n≥k ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ s✲ lim inf An , ✇✲ lim sup An ✈➔ LiAn , LsAn ❧➔ ♥❤÷ s❛✉ n→∞ n→∞ ∞ LiAn ⊂ s✲ lim inf An = n→∞ ✈➔ An ; k LiU k=1 ∞ LsAn ⊂ ✇✲ lim An = n→∞ ∞ n→∞ ∞ = U k=1 n=1 m≥n Am ; k ∞ ✇✲❝❧ Am m=n n=1 ✸✳ ❈❤♦ {An} ⊂ Kc(X) ❛✮ ❑❤✐ ✤â ✇✲ lim sup An = ∞ ✇✲ lim sup(An pU ) n→∞ p=1 ❜✮ ◆➳✉ X∗ ❧➔ ❦❤↔ ❧② ❤♦➦❝ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ t❤➻ ✇✲ lim sup An = n→∞ ∞ ∞ ✇✲❝❧ ∞ (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 tỗ t {nk } ⊂ N ✈➔ {xk } n→∞ s❛♦ ❝❤♦ xk ∈ An (k ≥ 1) ✈➔ (W )xk →x✳ ❱➟② k sup xk X = α < ∞ k≥1 ✶✻ ❈❤♦ p0 ∈ N s❛♦ ❝❤♦ p0 > α✱ ❝❤♦ k ≥ 1, xk ∈ An k ∩ p0 U t❤➻ x ∈ ✇✲ lim sup(An ∩ p0 U ) n→∞ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❜✮ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭❜✮ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ p ≥ t❤➻ ✇✲ lim sup(An ∞ ✇✲❝❧ pU ) = n→∞ n=1 ∞ (Am pU ) m=n ❚❤➟t ✈➟②✱ tø X∗ ❧➔ ❦❤↔ ❧②✱ ✤➦❝ ❜✐➺t ♥➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ t❤➻ pU ❧➔ ❝♦♠✲ ♣❛❝t tr õ ữủ ố ợ tổổ ứ ♥❣❤➽❛ ❝õ❛ ✇✲ lim sup(An ∩ n→∞ pU ) t❛ ❝â x ∈ ✇✲ lim sup(An pU ) ✈ỵ✐ ♠å✐ n ≥ 1✳ ❚ø ✤â t❛ ❝â x ∈ ✇✲❝❧ n (Am pU ) m=n ỵ ❈❤♦ {An, A} ⊂ Kbc(X) ✶✮ ◆➳✉ (H)An −→ A t❤➻ (W )An −→ A ✷✮ ◆➳✉ S(x∗ , An )−→ A(x∗ , A) ✈➔ K ∗ ⊂ X∗ t❤➻ (H)An −→ A ❇ê ✤➲ ✷✳✶✳✼✳✶✳ x ∈ coA ⇔ X∗ , x ≤ S(x, A); ∀x∗ ∈ X∗ ✳ ✷✳ ❈❤♦ {A, An, n ∈ N} ⊂ Kc(X) ✈➔ An ❤ë✐ tư tỵ✐ A t❤❡♦ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢✳ ❑❤✐ ✤â lim S(x∗ , An ) = S(x∗ , A), ∀x∗ ∈ X∗ n→∞ ✈ỵ✐ S(x∗ , A) = sup x∗ , a x X aA ỵ ✶✳ ◆➳✉ {An , A} ⊂ Kc (X) ✈➔ (H)An →A t❤➻ (KM )An →A✳ ✷✳ ◆➳✉ dim X < ∞, {An, A} ⊂ Kk (X) ✈➔ (KM )An →A t❤➻ (H)An →A ✶✼ ❈❤ù♥❣ ♠✐♥❤✳ ✶✳ ❈❤♦ x ∈ A s r tỗ t xn An ợ n ∈ N s❛♦ ❝❤♦ x − xn X < d(x, An ) + n ứ ỵ t❛ ❝â H(An, A) → s✉② r❛ d(x, An)→d(x, A) = 0✳ ❱ỵ✐ ♠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 r tỗ t n1 < n2 < ✈➔ xn ∈ An s❛♦ n→∞ ❝❤♦ (W )xn →x✳ ❚ø ❇ê ✤➲ ✷✳✶✳✼ t❛ ❝â x∗, xn ≤ S(x∗, An ) ✈ỵ✐ ♠å✐ x∗ ∈ X∗ ▼➦t ❦❤→❝✱ tø ❣✐↔ t❤✉②➳t (H)An→A ✈➔ ❇ê ✤➲ ✷✳✶✳✼ t❛ ❝â i i i i i lim S(x∗ , Ani ) = S(x∗ , A) n→∞ ✈ỵ✐ ♠å✐ x∗ ∈ X∗ ❱➻ ✈➟② t❛ ❝â (x∗, xn) ≤ S(x∗, An)✳ ❉♦ ✤â ✈ỵ✐ ♠å✐ x ∈ A ✈➔ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✼✳ ❙✉② r❛ (KM )An = A ✣à♥❤ ỵ {An, A} Kkc(x) dim X < ∞ t❤➻ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✶✳ (H)An→A ✷✳ (W ijs)An→A ✸✳ (KM )An→A ✹✳ (W )AnA ứ ỵ t s r ⇔ ✭✸✮✳ ❜✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✷✮ ⇒ ✭✸✮✳ ●✐↔ sû (W ijs)An → A ✈ỵ✐ x ∈ A t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ lim d(x, An ) = d(x, A) = n→∞ ❱ỵ✐ n ∈ N ✈➔ xn ∈ An t❤➻ x − xn X ≤ d(x, An ) + ✶✽ n s✉② r❛ s✲ n→∞ lim xn → x ✈ỵ✐ x ∈ s✲ lim inf An ✳ x lim sup An t tỗ t xk ∈ An ✈ỵ✐ ♠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 ) = k→∞ ❉♦ ✤â t❛ ❝â k k d(x, A) = lim d(x, An ) = lim d(x, Ank ) = n→∞ k→∞ s✉② r❛ x ∈ A✳ ❱➟② ✇✲ n→∞ lim sup An ⊂ A ⊂ lim inf An n→∞ ❝✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✶✮ ⇔ ✭✹✮✿ ✐✮ ứ ỵ t õ (H)An A t❤➻ (W )AN → A✳ ✐✐✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✹✮ ⇒ ✭✶✮✳ ●✐↔ sû lim sup sup |S(c∗ , An ) − S(x∗ , A)| > n→∞ x∗ ∈S r tỗ t n(1) < n(2) < ✈➔ x∗i ∈ S ∗ s❛♦ ❝❤♦ lim |S(x∗i , An(i) ) − S(x∗i , A)| > S t tr X s r tỗ t↕✐ ❞➣② i1 < i2 < ✈➔ x∗0 ∈ S ∗ s❛♦ ❝❤♦ lim x∗ij − x∗0 → j→∞ ▼➦t ❦❤→❝✱ tø ✤à♥❤ ♥❣❤➽❛ H(A, B) = max 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 nN x S tỗ t K Kkc(X) s❛♦ ❝❤♦ An ⊂ K ✈ỵ✐ ♠å✐ n ∈ N✳ ✶✾ ❚❛ ❝â ✈ỵ✐ ♠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→∞ lim sup |S(x∗0 , An(i) − S(x∗0 , A)| > ✭tr→✐ ✈ỵ✐ ❣✐↔ t❤✐➳t✮✳ (W )An A ỵ sỷ {An} ✈➔ {Bn} ❧➔ ❤❛✐ ❞➣② t❤✉ë❝ Kc(X) ✶✮ ♥➳✉ A ∈ Kc(X) ✈➔ lim sup s(x∗, An) ≤ s(x∗, A) ✈ỵ✐ ♠å✐ x∗ n→∞ ∈ X∗ t❤➻ ✇✲lim sup An ⊂ A n→∞ ✷✮ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ ❤♦➦❝ ❦❤æ♥❣ ❣✐❛♥ ❦❤↔ ❧②✳ ❑❤✐ ✤â ❛✮ ♥➳✉ supn An K < ∞ t❤➻ ✇✲lim sup An ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t ②➳✉ ❦❤→❝ n→∞ ∗ ∗ ré♥❣ ✈➔ lim sup s(x , An ) ≤ s(x , ✇✲ lim sup An ) ✈ỵ✐ x∗ ∈ X∗ n→∞ ❜✮ ♥➳✉ supn An K n→∞ < ∞ t❤➻ ✇✲ lim sup cl(An + Bn) ⊂ ✇✲ lim sup An + ✇✲ lim sup Bn n→∞ ❝✮ ♥➳✉ 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 tỗ t {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→∞ n→∞ ✈ỵ✐ ♠å✐ x∗ ∈ X∗ ❑➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✷✳✶✳✼ s✉② r❛ x ∈ A✳ ✷ ❛✮ ▲➜② r = supn An K < ∞✳ ❱➻ {x ∈ X : ♠❡tr✐❝ ❤â❛ ✤÷đ❝ t❤❡♦ tỉ♣ỉ ②➳✉✱ ♥➯♥ ✇✲ lim sup An = n→∞ ∞ ✇✲❝❧ m=1 ✷✵ x X ≤ r} ∞ An = ∅ n=m ❧➔ ❝♦♠♣❛❝t ✈➔ ❱ỵ✐ t❤➻ t❛ ❝â t❤➸ ❝❤å♥ ✤÷đ❝ ♠ët ❞➣② {xk : xk x∗ , xk → lim sup s(x∗ , An ) ✈➔ ✇✲ lim xk = x ✈ỵ✐ x ∈ X✳ k→∞ n→∞ ❉♦ ✤â 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 ✳ ❈❤ó♥❣ t❛ ❣✐↔ t❤✐➳t r➡♥❣ k→∞ lim xk = x ✈➔ ✇✲ lim yk = y = z − x✳ ❉♦ ✤â x→∞ k k z = x + y ∈ ✇✲ lim sup An + ✇✲ lim sup Bn n→∞ n→∞ ✷ ❝✮ ●✐↔ sû lim sup H(An, Bn) < ∞✳ ▲➜② x lim sup An t tỗ t n n→∞ {xk ∈ An } s❛♦ ❝❤♦ ✇✲ lim xk = x k→∞ ❱ỵ✐ ♠é✐ k ≥ ❝❤ó♥❣ t❛ ❝❤å♥ yk ∈ Bn s❛♦ ❝❤♦ k k xk − yk X ≤ H(Ank , Bnk ) + k ❉♦ ❞➣② {yk } ❜à ❝❤➦♥ ②➳✉ ♥➯♥ ✇✲k→∞ lim yk = y t❤✉ë❝ ✈➔♦ ✇✲ lim Bn ✈➔ n→∞ d(x, ✇✲ lim sup Bn ) ≤ x − y n→∞ X ≤ lim sup xk − yk k→∞ X ≤ lim sup H(An , Bn ) n→∞ ✣à♥❤ ỵ {An} Kk (X) ✈➔ ∞ A= An n=1 ❈❤ù♥❣ ♠✐♥❤✳ t❤➻ n→∞ lim H(An , A) = ợ > tỗ t↕✐ n ∈ N s❛♦ ❝❤♦ An ⊂ ∪(A; ε) ✈ỵ✐ ∪(A; ε) = {x ∈ x : d(x, A) < ε} ❚ø ❣✐↔ t❤✐➳t A = ∞ An n=1 t❛ ❝â Ac = ∞ n=1 ✷✶ Acn ✭♣❤➛♥ ❜ò✮✳ ❱➻ ✈➟② ∞ X = ∪(A; ε) ∪ Acn Acn n=1 ●✐↔ sû ∞ A1 ⊂ ∪(A; ε) ∪ n=1 ❱➻ A1 ❧➔ ❝♦♠♣❛❝t ✈➔ An ❧➔ ❞➣② s r tỗ t n N s A1 ⊂ ∪A(A; ε) ∪ Acn ✳ ❚ø ✤➙② s✉② r❛ A1 ∩ ∪(A; ε)c ∩ An = ∅ ❱➻ An ⊂ A1 t❤❡♦ ❣✐↔ t❤✐➳t ♥➯♥ ∪(A; ε)c ∩ An = ∅ s✉② r❛ An ⊂ ∪(An; ε) ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷✳ ❈⑩❈ ❉❸◆● ❍❐■ ❚Ö ❈Õ❆ ❇■➌◆ ◆●❼❯ ◆❍■➊◆ ✣❆ ❚❘➚ ●✐↔ sû (Ω, F, P) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✤➛② ✤õ✳ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❦❤↔ ❧✐✱ G ❧➔ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ F, B(E) ❧➔ σ✲✤↕✐ sè ❇♦r❡❧✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❚❛ ♥â✐ →♥❤ ①↕ X : Ω−→ E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✱ ♥❤➟♥ ❣✐→ trà tr♦♥❣ E ♥➳✉ X ❧➔ G/B(E) ✤♦ ✤÷đ❝ ✭♥❣❤➽❛ ❧➔ ✈ỵ✐ ♠å✐ B ∈ B(E) t❤➻ X −1 (B) ∈ G) P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ F ✲✤♦ ✤÷đ❝ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✷✳ ✶✳ ❉➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ (Xn) ❣å✐ ❧➔ ❤ë✐ tö ✤➳♥ →♥❤ ①↕ X : Ω−→ E ♥➳✉ Xn(ω)−→ X(ω) ✭t❤❡♦ ợ ỵ Xn −→ X ✷✳ ❉➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ (Xn) ❣å✐ ❧➔ ❤ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✭❤✳❝✳❝✮ ✤➳♥ →♥❤ ①↕ X : E tỗ t t N F s❛♦ ❝❤♦ P(N ) = 0, Xn(ω)−→X(ω) h.c.c ✭t❤❡♦ ợ \N ỵ Xn −→ X✳ ✷✷ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✸✳ ●✐↔ sû (Xn) ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ (Ω, F, P) ♥❤➟♥ ❣✐→ trà tr♦♥❣ (E, B(E)) ❚❛ ♥â✐ (Xn) ❤ë✐ tö ✤➳♥ X✿ ❤➛✉ ❝❤➢❝ ❝❤➢♥ ♥➳✉ P lim Xn − X = = n→∞ t❤❡♦ ①→❝ ①✉➜t ♥➳✉ ✈ỵ✐ ♠å✐ ε > t❤➻ lim P( Xn − X > ε) = n→∞ W ②➳✉ ✭t❤❡♦ ♣❤➙♥ ♣❤è✐✮ ♥➳✉ PX −→ PX tr♦♥❣ ✤â n PX : B(E)−→R B−→ P(X −1 (B)) h.c.c ❱➼ ❞ö ✷✳✷✳✹✳ ❈❤♦ Xn −→ ❈❤ù♥❣ ♠✐♥❤✳ X, Yn −→ Y ✳ h.c.c h.c.c ❑❤✐ ✤â Xn + Yn −→ ✣➦t Ω1 = ω : lim Xn (ω) − X(ω) = n→∞ Ω2 = ω : lim Yn (ω) − Y (ω) = n→∞ h.c.c ❚❤❡♦ ❣✐↔ t❤✐➳t Xn −→ X h.c.c ✈➔ Yn −→ Y ♥➯♥ P(Ω1 ) = P(Ω2 ) = ⇒ P(Ω1 ∩ Ω2 ) = ❑❤✐ ✤â ♥➳✉ ω = Ω1 ∩ Ω2 t❤➻   lim Xn (ω) − X(ω) = n→∞  lim Yn (ω) − Y (ω) = ⇔ n→∞  Xn (ω) −→ X(ω)  Yn (ω) −→ Y (ω) ▼➦t ❦❤→❝✱ ❞♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➯♥ Xn (ω) + Yn (ω) −→ X(ω) + Y (ω) ⇒ ω ω : lim Xn + Yn − X − Y (ω) = n→∞ ❉♦ ✤â Ω1 ∩ Ω2 ⊂ ω : lim Xn + Yn − X − Y (ω) = n→∞ ✷✸ X +Y✳ ♥➯♥ P (lim Xn + Yn − X − Y (ω) = 0) = ❱➟② h.c.c Xn + Yn −→ X + Y ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ h.c.c ỵ Xn X ❦❤✐ ✈ỵ✐ ♠å✐ ε > lim P sup Xm − X > ε = n→∞ ❈❤ù♥❣ ♠✐♥❤✳ m≥n ❱ỵ✐ ♠é✐ ε > ✈➔ ♠é✐ n = 1, 2, ✤➦t ∞ sup Xm − X > ε = Dn (ε) = m≥n ( Xm − X > ε) m=n ❑❤✐ ✤â Dn(ε) ❧➔ ❣✐↔♠ ❦❤✐ n t➠♥❣ ✈➔ ∞ Dnc (ε) ( Xm − X ≤ ε) = m=n ❑❤✐ ✤â ω∈ lim Xn − X = ⇔ lim Xn (ω) − X(ω) = n→∞ n→∞ ∀ε > 0, ∃n : Xm (ω) − X(ω) ≤ ε ∀m ≥ n ∀k, ∃n : Xm (ω) − X(ω) ≤ ∀m ≥ n k ∀k, ∃n : ω ∈ Dnc k ∞ ∞ k=1 n=1 ❉♦ ✤â ∞ ∞ Dnc lim Xn − X = = n→∞ k=1 n=1 ♥➯♥ h.c.c Xn −→ X ⇔ P k Dnc ⇔ω∈ k lim Xn − X = = n→∞ ✷✹ ∞ ∞ ⇔P ⇔P ⇔P k=1 n=1 ∞ Dnc n=1 ∞ Dnc n=1 Dnc k =1 k = ∀k = 1, 2, k = ∀k = 1, 2, ⇔ lim P(Dn (ε)) = ✈➻ Dn(ε) ❧➔ ❣✐↔♠✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✷✳✻✳ ❚ø ✤à♥❤ ♥❣❤➽❛ sü ❤ë✐ tư tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ K(X) t❛ s✉② r❛ r➡♥❣ ♥➳✉ {Fn} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❝→❝ t➟♣ ✤â♥❣✱ t❤➻ t❛ ❝â ✶✳ Fn ❤ë✐ tư ❤✳❝✳❝ tỵ✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà t➟♣ ✤â♥❣ F t❤❡♦ ♠❡tr✐❝ ❍❛✉s❞♦r❢❢ ♥➳✉ n→∞ H(Fn , F ) −→ h.c.c ◆❣❤➽❛ tỗ t t N P (N ) = s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ω ∈ N t❤➻ n→∞ H(Fn (ω), F (ω) −→ ✷✳ Fn ❤ë✐ tư t❤❡♦ ①→❝ s✉➜t tỵ✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà t➟♣ ✤â♥❣ F t❤❡♦ p ♥❣❤➽❛ ♠❡tr✐❝ ❍❛✉s❞♦r❢❢ ♥➳✉ H(Fn, F ) −→ ♥❣❤➽❛ ❧➔ ∀ε > t❤➻ n→∞ P(H(Fn , F ) > ε) −→ ú ỵ r t ụ õ tữỡ tü ❝❤♦ sü ❤ë✐ tö ✤➛② ✤õ✱ ❤ë✐ tö ♣❤➙♥ ♣❤è✐✱ ❤ë✐ tö ②➳✉ t❤❡♦ ♠❡tr✐❝ ❍❛✉s❞♦r❢❢✳ ✸✳ ❉➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà t➟♣ ✤â♥❣ ❤ë✐ tö ❤✳❝✳❝ t❤❡♦ ♥❣❤➽❛ ❑✉r❛st♦✇s❦✐✲▼♦s❝♦ tỵ✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà t➟♣ ✤â♥❣ F ♥➳✉ (KM )Fn −→F h.c.c, ♥❣❤➽❛ ❧➔ tỗ t t N p(N ) = s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ω ∈ N t❛ ❝â (KM )Fn ()F () ứ ỵ t s✉② r❛ H Fn −→ F h.c.c t❤➻ Fn −→ F KM ✷✻ h.c.c ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ❦➳t q✉↔ s❛✉ ✶✳ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ✈➲ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà ✤â♥❣✳ ✷✳ ❚r➻♥❤ ❜➔② ❝â ❤➺ t❤è♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ✈➲ sü ❤ë✐ tư ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ✤â♥❣✳ ✸✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ỵ sỹ tử t tr sr s ✈➔ ❲✐❥s♠❛♥✳ ❚ø ✤â ❝❤➾ r❛ sü t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❝→❝ ❞↕♥❣ ❤ë✐ tư ♥➔② tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ✤â♥❣✳ ✹✳ ❚ø sü ❤ë✐ tư tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ K(X) ❞➝♥ ✤➳♥ ❦➳t ❧✉➟♥ ✈➲ sü ❤ë✐ tö ❝õ❛ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà ✤â♥❣ t❤❡♦ ♥❣❤➽❛ ♠❡tr✐❝ sr st s ữợ ự t t ◆❣❤✐➯♥ ❝ù✉ ✈➲ ❧✉➟t sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤❛ trà ✈ỵ✐ ❝→❝ ❞↕♥❣ ❤ë✐ tư ❦❤→❝ ♥❤❛✉✳ ✷✼ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ✣➟✉ ❚❤➳ ❈➜♣ ✭✷✵✵✵✮✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆①❜ ●✐→♦ ❞ö❝✱ ❍➔ ◆ë✐✳ ❬✷❪ ◆❣✉②➵♥ ❳✉➙♥ ▲✐➯♠ ✭✶✾✾✻✮✱ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣✱ ◆①❜ ●✐→♦ ❞ö❝✱ ❍➔ ◆ë✐✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣ ✭✷✵✵✽✮✱ ❳→❝ s✉➜t ♥➙♥❣ ❝❛♦✱ ◆①❜ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✹❪ ■❧②❛ ▼♦❧❝❤❛♥♦✈ ✭✷✵✵✺✮✱ ❚❤❡♦r② ♦❢ ❘❛♥❞♦♠ s❡ts✱ ❙♣r✐♥❣❡r✱ ▲♦♥❞♦♥✳ ❬✺❪ ❙✳ ▲✐✱ ❨✳ ❖❣✉r❛✱ ❱✳ ❑r❡✐♥♦✈✐❝❤ ✭✷✵✵✷✮✱ ▲✐♠✐t t❤❡♦r❡♠s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ s❡t✲✈❛❧✉❡❞ ❛♥❞ ❢✉③③② s❡t✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ❦❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜✲ ❧✐s❤❡rs ●r♦✉♣✱ ❉♦r❞r❡❝❤t✳ ✷✽ ... (K(X, H) ❦❤↔ ❧②✳ ▲➜② ♠ët t➟♣ ❝♦♥ D ✤➳♠ ✤÷đ❝ trị ♠➟t tr♦♥❣ X✳ ●å✐ D ❧➔ t➟♣ ❝→❝ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ D✳ ❑❤✐ ✤â D ❧➔ ✤➳♠ ✤÷đ❝✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ D ❧➔ trị ♠➟t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ K(X)✳ l ❱ỵ✐ ộ E K(X)... tỉ✐ q✉②➳t ✤à♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✏❈→❝ ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤❛ tr ữủ tỹ ữợ sỹ ữợ t➟♥ t➻♥❤ ✈➔ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ P●❙✳❚❙✳ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉... ✈➲ sü ❤ë✐ tö ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ t➟♣ ✤â♥❣✳ ✸✳ ❈❤ù♥❣ tt ởt số ỵ sỹ tö t❤❡♦ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢✱ ▼♦s❝♦ ✈➔ ❲✐❥s♠❛♥✳ ❚ø ✤â ❝❤➾ r❛ sü t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❝→❝ ❞↕♥❣ ❤ë✐ tư ♥➔②