Đặc trưng của tập fractal có độ đo hausdoff dương

▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ❈→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ✺ ✶✳✶✳ ❈→❝ ❧♦↕✐ →♥❤ ①↕✱ ❤➺ ❤➔♠ ❧➦♣✱ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ✈➔ ♠ð ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ t t tỹ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸✳ ✣ë ✤♦ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹✳ ✣ë ✤♦ ❍❛✉s❞♦❢❢ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❈❤÷ì♥❣ ✷✳ ✣➦❝ tr÷♥❣ ❝õ❛ t➟♣ ❢r❛❝t❛❧ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳ ▼ët sè ✈➼ ❞ö ✈➲ t➟♣ ❢r❛❝t❛❧ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳ ▼èt sè t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ❢r❛❝t❛❧ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳ ✣➦❝ tr÷♥❣ ❝õ❛ t tỹ ỗ õ sr ữỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶ ▼Ð ✣❺❯ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ❝ị♥❣ ✈ỵ✐ ❝❤✐➲✉ ❍❛✉s❞♦r❢❢ ❧➔ ❤❛✐ ✈➜♥ ✤➲ ❝ì ❜↔♥ ❝õ❛ ❤➻♥❤ ❤å❝ ❢r❛❝t❛❧✳ ▼ët ✤➦❝ ✤✐➸♠ ❝õ❛ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ Hs ❧➔ ✈ỵ✐ ♠é✐ t➟♣ F tr Rn trữợ t Hs (F ) tở tr s ổ tỗ t ♥❤➜t ♠ët ❣✐→ trà sF ♠➔ ✈ỵ✐ s < sF t❤➻ Hs (F ) = ∞✱ ❝á♥ ✈ỵ✐ s > sF t❤➻ Hs (F ) = 0✳ ❚↕✐ s = sF t❤➻ ❝â t❤➸ Hs (F ) = ∞✱ ❤♦➦❝ Hs (F ) = 0✱ ❤♦➦❝ < Hs (F ) < ∞ ❙è sF ♥❤÷ ✈➟②✱ ♥❣÷í✐ t❛ ❣å✐ ❧➔ ❝❤✐➲✉ ❍❛✉s❞♦❢❢ ❝õ❛ F ✳ ◆➠♠ ✷✵✵✹✱ t↕✐ ❤ë✐ ♥❣❤à t♦→♥ ❤å❝ ✈➲ ❤➻♥❤ ❤å❝ ❢r❛❝t❛❧✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ✤➲ ❝➟♣ ✤➳♥ ✶✷ ❜➔✐ t♦→♥ ✤è✐ ✈ỵ✐ ✤ë ❞♦ ❍❛✉s❞♦r❢❢✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❜➔✐ t♦→♥ ✤â ❧➔ ợ trữ t rt t t s = sF ✤ë ✤♦ ❍❛✉s❞♦r❢❢ s✲❝❤✐➲✉ Hs ♥❤➟♥ ❣✐→ trà ❞÷ì♥❣✱ ❤ú✉ ❤↕♥ ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ ❧♦↕✐ t➟♣ ♥➔② ❧➔ ❣➻❄ ✣➙② ❧➔ ♠ët ❜➔✐ t♦→♥ t❤ó ✈à t❤✉ ❤ót sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tø ✤➛✉ ♥❤ú♥❣ ♥➠♠ ✽✵ ❝õ❛ t❤➳ ❦✛ ✷✵✳ rữợ t ts tr ❤ä✐ ♥➔② ❝❤♦ tr÷í♥❣ ❤đ♣ t➟♣ ❢r❛❝t❛❧ ❧➔ t➟♣ tü ỗ t r ự t ởt ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët t➟♣ ❝â ✤ë ✤♦ ❞÷ì♥❣✱ ❤ú✉ ❤↕♥ ❝❤♦ t➟♣ tü ❜↔♦ ❣✐→❝✳ ❱✐➺❝ ✤→♥❤ ❣✐→ ❣✐→ trà ❝õ❛ ♠ët t➟♣ ❝â ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❞÷ì♥❣ ❤❛② ❦❤ỉ♥❣ s➩ ❣✐ó♣ ❝❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ ✈➲ sr ỗ tớ õ ú ú t t tử ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤ë ✤♦ ❝❤♦ ❝→❝ ❧ỵ♣ ❝→❝ t➟♣ ❢r❛❝t❛❧ rë♥❣ ❤ì♥ ✈➔ t➻♠ ❤✐➸✉ ✤÷đ❝ ♥❤✐➲✉ ❤ì♥ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â✳ ❱ỵ✐ ❧➼ ❞♦ ✤â ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✷ ❧➔ ✣➦❝ tr÷♥❣ ❝õ❛ t➟♣ ❢r❛❝t❛❧ ❝â ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❞÷ì♥❣✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t❤æ♥❣ q✉❛ ❝→❝ t➔✐ ❧✐➺✉ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ✈➲ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ✈➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ t➟♣ ❝â ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❞÷ì♥❣✳ ❚➻♠ ❦✐➳♠ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ✤➦❝ tr÷♥❣ ✤â✳ ❚❤ỉ♥❣ q✉❛ ✈✐➺❝ ❧➔♠ ❧✉➟♥ ✈➠♥ ❣✐ó♣ ❡♠ t➟♣ ❞✉②➺t ✈➔ ❧➔♠ q✉❡♥ ✈ỵ✐ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ ❝â ❤✐➸✉ ❜✐➳t ❤ì♥ ✈➲ ❧➽♥❤ ✈ü❝ ♠➻♥❤ ♥❣❤✐➯♥ ❝ù✉✳ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ♥ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ tr♦♥❣ ✷ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝ì sð ✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ❝→❝ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❤÷ t➟♣ ❢r❛❝t❛❧✱ ❤➺ ❤➔♠ ❧➦♣✱ t➟♣ t t tỹ ỗ ❍❛✉s❞♦r❢❢✱ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð✱ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ♠↕♥❤✱ ✤✐➲✉ ❦✐➺♥ ❇❛♥❞t✲●r❛❢✳ ❈❤÷ì♥❣ ✷✳ ✣➦❝ tr÷♥❣ ❝õ❛ t➟♣ ❢r❛❝t❛❧ ❝â ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❞÷ì♥❣✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ t➟♣ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ữợ sỹ ữợ ổ ụ ỗ tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t tỵ✐ ❝ỉ✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ❝õ❛ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ✸ t➟♥ t➻♥❤ ❞↕② ❞é✱ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ❣û✐ ỡ t t qỵ ổ ❣✐→♦ ❦❤♦❛ ❚♦→♥✱ ❦❤♦❛ ❙❛✉ ✤↕✐ ❤å❝ ✲ ❚r÷í♥❣ ✣↕✐ ỗ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ị t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ♥➠♥❣ ❧ü❝✱ ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ❝→❝ t❤✐➳✉ sõt t ữủ ỵ õ õ qỵ t ổ ữủ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ◆❣❤➺ ❆♥✱ ♥❣➔② ✵✾ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ❣✐↔ ✹ ❈❍×❒◆● ✶ ❈⑩❈ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ ❝→❝ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❤÷ t➟♣ ❢r❛❝t❛❧✱ ❤➺ ❤➔♠ ❧➦♣✱ t➟♣ ❜➜t ❜✐➳♥✱ t➟♣ tỹ ỗ sr t➟♣ ♠ð✱ ✤✐➲✉ ❦✐➺♥ ❇❛♥❞t✲●r❛❢✳ ✶✳✶ ❈→❝ ❧♦↕✐ →♥❤ ①↕✱ ❤➺ ❤➔♠ ❧➦♣✱ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ✈➔ ♠ð ♠↕♥❤ r ỵ |x y| ữủ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ t❤ỉ♥❣ t❤÷í♥❣ ❣✐ú❛ ❤❛✐ ♣❤➛♥ tû x ✈➔ y tr♦♥❣ Rn , ✈➔ ❉ ❧➔ t➟♣ ❝♦♥ ✤â♥❣✱ ❦❤→❝ ré♥❣ tr♦♥❣ Rn ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ →♥❤ ①↕ S : D→D ✐✮ ❙ ✤÷đ❝ ❣å✐ tr D tỗ t c ∈ [0; 1) ✤➸ |S(x) − S(y)| ≤ c|x − y|, ∀x, y ∈ D ✈➔ c ✤÷đ❝ ❣å✐ ❧➔ t✛ sè ❝♦ ❝õ❛ →♥❤ ①↕ S ✳ ✐✐✮ ◆➳✉ ❞➜✉ ✧❂✧ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❧✉æ♥ ①➞② r❛✱ ♥❣❤➽❛ ❧➔ |S(x) − S(y)| = c|x − y|, ∀x, y ∈ D, c ∈ [0, 1) t❤➻ ❙ ✤÷đ❝ ữủ ỗ c ữủ t số ỗ ①↕ S ✳ ✐✐✐✮ S ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ st tr D tỗ t c > |S(x) − S(y)| ≤ c|x − y|, ∀x, y ∈ D ✐✈✮ S ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ s♦♥❣ ▲✐♣s❤✐t③ tr D tỗ t c1 , c2 > ✤➸ c1 |x − y| ≤ |S(x) − S(y)| ≤ c2 |x − y|, ∀x, y ∈ D ✈✮ S ữủ r tỗ t sè c > ✈➔ α > ✤➸ |S(x) − S(y)| ≤ c|x − y|α , ✺ x, y ∈ D ✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ▼ët ❤å ❤ú✉ ỗ m {S1, S2, , Sm} tr➯♥ D ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ❤➔♠ ❧➦♣ ✭✈✐➳t t➢t ❧➔ ■❋❙ ✲ ■t❡r❛t❡❞ ❋✉♥❝t✐♦♥ ❙②st❡♠✮ tr➯♥ D✳ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❚❛ ♥â✐ r➡♥❣ ❤➺ ❤➔♠ ❧➦♣ {S1, S2, , Sm} tr➯♥ D t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ✭✈✐➳t t➢t ❧➔ ❖❙❈✲ t t tỗ t t V ré♥❣ tr♦♥❣ Rn ✤➸  V ⊃ m S (V ) i i=1  Si (V ) ∩ Sj (V ) = ∅ ∀i = j ✭✶✳✶✮ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❚❛ ♥â✐ r➡♥❣ ❤➺ ❤➔♠ ❧➦♣ {S1, S2, , Sm} tr➯♥ D t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ♠↕♥❤ ✭✈✐➳t t➢t ❧➔ ❙❖❙❈✲ ❙tr♦♥❣ ❖♣❡♥ ❙❡t ❈♦♥❞✐t✐♦♥✮ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ✈➔ t❤➯♠ ✤✐➲✉ ❦✐➺♥ m Si (V ) = ∅ F∩ i=1 ✈ỵ✐ F ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ❤➺ ❤➔♠ õ t t tỹ ỗ ❈❤♦ U ⊆ Rn , U = ∅✳ ❑❤✐ ✤â✱ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ t➟♣ U ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ |U | = sup{|x − y| : x, y ∈ U } ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ A ❧➔ ♠ët t➟♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ (X, d)✳ ✐✮ ❱ỵ✐ ♠é✐ ✤✐➸♠ x ∈ X t❛ ✤➦t d(x, A) = inf{d(x, y) : y ∈ A} ✈➔ t❛ ❣å✐ d(x, A) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø x ✤➳♥ t➟♣ A✳ ✐✐✮ ❈❤♦ δ > 0✱ t➟♣ Aδ = {x ∈ X : d(x, A) } ỗ ỳ t A ♠ët ❦❤♦↔♥❣ ❦❤æ♥❣ q✉→ δ ✱ ❣å✐ ❧➔ δ ✲❜❛♦ A ỵ t ❝♦♠♣❛❝t ❦❤→❝ ré♥❣ tr♦♥❣ Rn✳ ❑➼ ❤✐➺✉ K ❧➔ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t✱ ❦❤→❝ ré♥❣ ❝õ❛ ❉✳ ❑❤✐ ✤â✱ ❤➔♠ dH : K × K→R (A, B) → dH (A, B) = inf{δ ≥ : A ⊆ Bδ , B ⊆ Aδ } t❤ä❛ ♠➣♥ ✐✮ dH (A, B) = max sup d(x, B); sup d(y, A) ✳ x∈A y∈B ✐✐✮ dH ❧➔ ♠ët ♠➯tr✐❝ tr➯♥ K✳ ❍ì♥ ♥ú❛✱ ❦❤æ♥❣ ❣✐❛♥ (K, dH ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✳ ✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ▼➯tr✐❝ dH tr➯♥ K tr ỵ ữủ tr sr tr➯♥ K✳ ✶✳✷✳✹ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ❈❤♦ ♠ →♥❤ ①↕ ❝♦ {Si}mi=1 tr➯♥ ❉✳ ❚❛ ①→❝ ✤à♥❤ →♥❤ ①↕ S : K→K ❜ð✐ m E → S(E) = Si (E) ✭✶✳✷✮ i=1 t❤➻ dH (S(A), S(B)) ≤ cmax dH (A, B) ✈ỵ✐ cmax = max {ci }✱ ci ❧➔ t➾ sè ❝♦ ❝õ❛ 1≤i≤m →♥❤ ①↕ Si , i = 1, , m✳ ❉♦ ✤â✱ S ❧➔ →♥❤ ①↕ ❝♦ ✈ỵ✐ t➾ sè ❝♦ ❧➔ cmax ✳ ❚ø ✣à♥❤ ❧➼ ✶✳✷✳✷ ✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✹✱ t❤❡♦ ♥❣✉②➯♥ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ t❛ ❝â ✣à♥❤ ❧➼ s❛✉✳ ✶✳✷✳✺ ỵ {Si}mi=1 S ữủ õ tỗ t↕✐ ❞✉② ♥❤➜t ♠ët t➟♣ F ∈ K s❛♦ ❝❤♦ S(F ) = F ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ ❝â t➟♣ E ∈ K s❛♦ ❝❤♦ Si (E) ⊆ E, ≤ i ≤ m t❤➻ ∞ S k (E) F = k=1 ✈ỵ✐ S k ❧➔ sü ❧➦♣ ❧↕✐ k ❧➛♥ ❝õ❛ →♥❤ ①↕ S (k = 1, 2, )✳ ✼ ✶✳✷✳✻ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ✐✮ ❈❤♦ ❤➺ ❤➔♠ ❧➦♣ {Si}mi=1 tr➯♥ D✳ ❚➟♣ F ✤÷đ❝ ①→❝ ữ tr ỵ ữủ t ❜➜t ❜✐➳♥ ✭❛tr❛❝t♦r s❡t✮ ❤❛② t➟♣ ❤ót ❝õ❛ ❤➺ ❤➔♠ ❧➦♣ {Si }m i=1 ✳ ✐✐✮ ◆➳✉ ❝→❝ Si (1 i m) ỗ t❤➻ t➟♣ ❜➜t ❜✐➳♥ F ✤÷đ❝ ❣å✐ ❧➔ t➟♣ tü ỗ t t ữủ ❝→❝ t➟♣ ❢r❛❝t❛❧✳ ▼ët t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ t➟♣ rt t tỹ ỗ õ ởt ọ tũ ỵ õ ởt t tỹ ỗ F t ữủ ❧✉æ♥ ❧➔ ✧❜↔♥ s❛♦✧ ❝õ❛ F ✳ ✶✳✸ ✣ë ✤♦ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✶✳✸✳✶ ✣à♥❤ ♥❣❤➽❛ X ởt t ủ tũ ỵ ✈➔ C ❧➔ ✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X t : C R ữủ ❧➔ ♠ët ✤ë ✤♦ tr➯♥ C ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ s à(A) ợ A ∈ C; ✐✐✮ µ(∅) = 0; ✐✐✐✮ µ ❧➔ σ−❝ë♥❣ t➼♥❤✱ tù❝ ❧➔ ♥➳✉ ∞ Ai ∈ C(i = 1, 2, ), Ai ∩ Aj = ∅(i = j), Ai ∈ C i=1 t❤➻ ∞ µ( ∞ Ai ) = i=1 µ(Ai ) i=1 ✷✳ ✣ë ✤♦ µ tr➯♥ σ−✤↕✐ sè C ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤õ ♥➳✉ ♠å✐ t➟♣ ❝♦♥ ❝õ❛ ♠ët t➟♣ ❜➜t ❦ý t❤✉ë❝ C ❝â ✤ë ✤♦ ✵ ✤➲✉ t❤✉ë❝ C ✈➔ ❝â ✤ë ✤♦ ✵✱ ♥❣❤➽❛ ❧➔ ♥➳✉ A ⊂ B, B ∈ C, µ(B) = t❤➻ A ∈ C à(A) = ữủ ✤ë ✤♦ ♥❣♦➔✐ tr➯♥ C ♥➳✉ t❤ä❛ ♠➣♥ ✐✮✱ ✐✐✮ t ữợ t➼♥❤✱ tù❝ ❧➔ ♥➳✉ ∞ Ai ∈ C (i = 1, 2, ), Ai ∈ C i=1 t❤➻ ∞ ∞ Ai ) ≤ µ( i=1 µ(Ai ) i=1 ✹✳ ✣ë ✤♦ µ ❧➔ ❤ú✉ ❤↕♥ ♥➳✉ µ(X) < +∞✳ ✶✳✸✳✷ ✣à♥❤ ❧➼ ✭❬✸❪✮✳ ◆➳✉ µ ❧➔ ✤ë ✤♦ tr➯♥ ✤↕✐ sè C t❤➻ t❛ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✐✮ ◆➳✉ A, B ∈ C, A ⊂ B t❤➻ µ(A) ≤ µ(B);✳ ✐✐✮ ◆➳✉ µ(B) < +∞ t❤➻ µ(A\B) = µ(A) − µ(B);✳ B⊂A ✐✐✐✮ ◆➳✉ A, Ai ∈ C, A ⊂ ∞ Ai t❤➻ µ(A) ≤ k=1 ✐✈✮ ◆➳✉ A, Ai ∈ C, Ai ∩ Aj = ∅, A ⊃ ∞ µ(Ai ); k=1 ∞ Ai t❤➻ µ(A) ≥ k=1 ✶✳✸✳✸ ❍➺ q✉↔ ✭❬✸❪✮✳ ◆➳✉ ∞µ ❧➔ ✤ë ✤♦ tr➯♥ ✤↕✐ sè C t❤➻ ✐✮ ◆➳✉ µ(Ai ) = 0, ∀i ✈➔ ∞ µ(Ai ) k=1 Ai ∈ C t❤➻ µ(∪Ai ) = k=1 ✐✐✮ ◆➳✉ A ∈ C, µ(B) = t❤➻ µ(A ∪ B) = µ(A \ B) = µ(A)✳ ✶✳✸✳✹ ❍➺ q✉↔ ✭❬✸❪✮✳ ❈❤♦ µ ❧➔ ∞✤ë ✤♦ tr➯♥ ✤↕✐ sè (C) t❛ ❝â ∞ Ai t❤➻ µ( ✐✮ ◆➳✉ Ai ∈ C, A1 ⊂ A2 ⊂ i=1 ∞ ✐✐✮ ◆➳✉ Ai ∈ C, A1 ⊃ A2 ⊃ Ai ) = lim µ(Ai ) i=1 ∞ Ai t❤➻ µ( i=1 i=1 i→∞ Ai ) = lim µ(Ai ) i→∞ ✶✳✸✳✺ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ●✐↔ sû X = (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ❦❤✐ ✤â✱ σ− ✤↕✐ sè ♥❤ä ♥❤➜t ❜❛♦ ❤➔♠ ❧ỵ♣ ❝→❝ t➟♣ ♠ð tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ✤÷đ❝ ❣å✐ ❧➔ σ− ✤↕✐ sè ❇♦r❡❧ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X ✈➔ ♥❤ú♥❣ t➟♣ t❤✉ë❝ σ− ✤↕✐ sè ♥➔② ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❇♦r❡❧ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ X ✶✳✸✳✻ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ●✐↔ sû µ ✈➔ ν ❧➔ ✤ë ✤♦ ①→❝ ✤à♥❤ tr➯♥ C ✳ ❑❤✐ ✤â✱ ν ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tử tt ố ố ợ (E) = 0, ✈ỵ✐ ♠å✐ E ∈ C ✾ t❤ä❛ ♠➣♥ µ(E) = 0, ❦➼ ❤✐➺✉ ❧➔ ν ◆➳✉ µ µ✳ µL t❛ ♥â✐ µ ❧➔ ❧➯♥ t✉❝ t✉②➺t ✤è✐ ✭µL −✤ë ✤♦ ▲❡❜❡s❣✉❡✮ ✶✳✸✳✼ ✣à♥❤ ❧➼✳ ✭❈❛r❛t❤❡♦❞♦r②✮ ✭❬✸❪✮✳ ●✐↔ sû µ∗ ❧➔ ✤ë ✤♦ ♥❣♦➔✐ tr➯♥ X ✳ ❑➼ ❤✐➺✉ L = {A ⊂ X : µ∗ (E) = µ∗ (E ∩ A) + µ∗ (E\A), ∀E ⊂ X} ❑❤✐ ✤â L ❧➔ ✤↕✐ sè ✈➔ µ = µ∗ L ã ữủ s ã A L ữủ ữủ ã A L ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ µ∗ (E) ≥ µ∗ (E ∩ A) + (E\A) ữủ ổ ú ✶✳✹ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✶✳✹✳✶ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ ∞{Ui}∞i=1 ❧➔ ♠ët ❤å ✤➳♠ ✤÷đ❝ ❝→❝ t➟♣ ❝♦♥ tr♦♥❣ Rn ✈➔ F ⊂ Rn ✳ ◆➳✉ F ⊂ i=1 Ui t❤➻ {Ui }∞ i=1 ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤õ ❝õ❛ F ◆➳✉ t❤➯♠ ✤✐➲✉ ❦✐➺♥ < |Ui | ≤ δ ✈ỵ✐ ♠å✐ i✱ tr♦♥❣ ✤â > trữợ t {Ui } i=1 ữủ ❣å✐ ❧➔ ♠ët δ ✲♣❤õ ❋✳ ❱ỵ✐ F ⊂ Rn , s ≥ ✈➔ δ > t❛ ✤➦t ∞ Hδs (F ) |Ui |s : {Ui } ❧➔ δ − ♣❤õ F = inf i=1 ❑❤✐ ✤â✱ ổ tỗ t lim Hs (F ) ũ ợ ✤â ❝â t❤➸ ❜➡♥❣ ❤❛② ∞✮✳ ✣✐➲✉ δ→o ♥➔② ❞➝♥ tỵ✐ ❦➳t q✉↔ s❛✉✳ ✶✳✹✳✷ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ❈❤♦ F ⊂ Rn, s ≥ 0✱ ✤➦t Hs(F ) = δ→0 lim Hδs (F ) t❤➻ Hs : P(Rn )→R ❧➔ ♠ët ✤ë ✤♦ ♥❣♦➔✐ tr➯♥ Rn ✈ỵ✐ P(Rn ) ❧➔ ❧ỵ♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ Rn ✳ ✶✵ ✷✳✷✳✸ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ●✐↔ sû ❋ ❧➔ t➟♣ ❇♦r❡❧ t❤ä❛ ♠➣♥ < Hs(F ) < +∞ ❑❤✐ õ tỗ t số b > t ❝♦♠♣❛❝t E ⊂ F ✈ỵ✐ Hs (E) > s❛♦ ❝❤♦ Hs (E ∩ B(x, r)) ≤ brs , ∀x ∈ Rn , r > ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t µ ❧➔ ✤ë ✤♦ t❤✉ ❤➭♣ ❝õ❛ Hs ❧➯♥ F ✱ ợ A Rn t õ à(A) = Hs (F ∩ A) ❑❤✐ ✤â✱ ♥➳✉ F1 = Hs (F ∩ B(x, r)) x ∈ R : lim > 21+s s r→0 r n t❤➻ Hs (F1 ) ≤ 2s 2−(1+s) µ(F ) = Hs (F ) ◆❤÷ ✈➟②✱ Hs (F \ F1 ) ≥ Hs (F ) > s ❉♦ ✤â✱ ♥➳✉ E1 = F \ F1 t❤➻ Hs (E1 ) > ✈➔ lim H (F ∩B(x,r)) ≤ 21+s ✈ỵ✐ x ∈ E1 rs r0 r t tỗ t↕✐ E ⊂ E1 ✈ỵ✐ Hs (E) > ✈➔ sè r0 > s❛♦ ❝❤♦ Hs (F ∩ B(x, r)) ≤ 22+s s r ✈ỵ✐ ∀x ∈ E ✈➔ ∀ < r < r0 ❚✉② ♥❤✐➯♥✱ t❛ ❝â Hs (F ∩ B(x, r)) Hs (F ) ≤ ♥➳✉ r > r0 rs r0s ❱➟②✱ Hs (E ∩ B(x, r)) ≤ brs , ∀x ∈ Rn , r > ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✶✽ ✷✳✷✳✹ ❇ê ✤➲ ✭❬✸❪✮✳ ✭❇ê ✤➲ ♣❤õ✮✳ ●✐↔ sû C ❧➔ ❤å ❝→❝ ❤➻♥❤ ❝➛✉ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ♠ët ♠✐➲♥ Rn õ tỗ t ❝♦♥ ✭❤ú✉ ❤↕♥ ❤♦➦❝ ✤✐➳♠ ✤÷đ❝✮ ❝→❝ ❤➻♥❤ ❝➛✉ rí✐ ♥❤❛✉ {Bi }s❛♦ ❝❤♦ B⊂ Bi B∈C ✈ỵ✐ Bi ❧➔ õ t ỗ t ợ Bi ữ ❜→♥ ❦➼♥❤ ❧ỵ♥ ❣➜♣ ✹ ❧➛♥✳ ✷✳✷✳✺ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ sỷ ởt sỹ ố ố ữủ tr➯♥ Rn, F ⊂ Rn ❧➔ t➟♣ ❇♦r❡❧ ✈➔ ❧➜② < c < ∞ ❧➔ ❤➡♥❣ sè✳ ❑❤✐ ✤â✱ ♥➳✉ µ(B(x, r)) < c ∀x ∈ F r→0 rs lim t❤➻ Hs (F ) ≥ µ(F ) > ✈ỵ✐ s = ❞✐♠H F c ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ δ > ❧➜② Fδ = {x ∈ F : µ(B(x, r)) < crs ∀r, < r < δ} ▲➜② {Ui } ❧➔ δ ✲♣❤õ F ✈➔ ❞♦ ✤â ♥â ♣❤õ Fδ ❱ỵ✐ ♠é✐ Ui ❝❤ù❛ ♠ët ✤✐➸♠ x ❝õ❛ Fδ , t❤➻ ❤➻♥❤ ❝➛✉ B ✈ỵ✐ t➙♠ x ✈➔ ❜→♥ ❦➼♥❤ |Ui | ❝❤ù❛ Ui ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Fδ ✱ t❛ ❝â µ(Ui ) ≤ µ(B) ≤ c|Ui |s ❉♦ ✤â✱ µ(Fδ ) ≤ |Ui |s {µ(Ui ) : Ui ∩ Fδ = ∅} ≤ c i i ❱➻ {Ui } ❧➔ δ ✲♣❤õ ❜➜t ❦➻ ❝õ❛ F, ♥➯♥ µ(Fδ ) ≤ cHδs (F ) ≤ cHs (F ) ◆❤÷♥❣ Fδ t✐➳♥ ✈➲ F ❦❤✐ δ → ♥➯♥ t❛ ❝â µ(F ) ≤ cHs (F ), ✶✾ ❤❛② Hs (F ) ≥ µ(F ) c > ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ❧➼ s❛✉ ✤➙② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝❤✐➲✉ ❝õ❛ ❝❤✉②➸♥ ✤ë♥❣ ❇r♦✇♥✐❛♥✳ ◆â ❝ơ♥❣ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ t➻♠ ❝❤✐➲✉ ❍❛✉s❞♦r❢❢ ❝õ❛ Fθ ❦❤✐ θ t❤❛② ✤ê✐✳ ▼ët ❝→❝❤ tü ♥❤✐➯♥ t❛ ố ố ữủ tr F ợ ♠é✐ θ✳ ◆➳✉ t❛ ❝â dµθ (x)dµθ (y)dθ 0} ỵ sỷ t Rn, s > tỗ t ởt ố ố ữủ tr ợ Is (µ) < ∞ t❤➻ Hs (F ) = ∞ ✈➔ ❞✐♠H F ≥ s ✐✐✮ ◆➳✉ F ❧➔ t➟♣ ❇♦r❡❧ ợ Hs (F ) > t tỗ t ởt ố ố ữủ tr F ợ It (à) < ∞, ∀ < t < s ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ Is (à) < ợ ố ố ữủ µ ♥➔♦ ✤â ❝â ❣✐→ µ ❧➔ s♣tµ ⊂ F ✣➦t F1 = µ(B(x, r)) >0 r→0 rs x ∈ F : lim ✷✵ ◆➳✉ x ∈ F1 ✱ t❛ t➻♠ ✤÷đ❝ ε > ✈➔ ❞➣② ❝→❝ sè ỳ t {ri } tợ s à(B(x, ri )) ≥ εris ❱➻ µ({x}) = ✭✈➻ ♥➳✉ ❦❤ỉ♥❣ t❤➻ Is (µ) = ∞✮ ♥➯♥ tø t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ µ, ✈➔ ❜➡♥❣ ✈✐➺❝ ❧➜② qi (0 < qi < ri ) ✤õ ♥❤ä t❛ ❝â µ(Ai ) ≥ 14 εris (i = 1, 2, ) ✈ỵ✐ Ai ❧➔ ❤➻♥❤ ✈➔♥❤ ❦❤✉②➯♥ B(x, ri ) \ B(x, qi ) ▲➜② ❞➣② ❝♦♥ ♥➳✉ ❝➛♥✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ ri+1 < qi ∀i ∈ N∗ ❉♦ ✤â✱ Ai ❧➔ ❝→❝ ❤➻♥❤ ✈➔♥❤ ❦❤✉②➯♥ rí✐ ♥❤❛✉ t➙♠ t↕✐ x✳ ❑❤✐ ✤â✱ ✈ỵ✐ ∀x ∈ F1 t❛ ❝â dµ(y) ≥ |x − y|s φs (x) = ∞ ≥ i=1 s −s εri ri ∞ i=1 Ai dµ(y) |x − y|s =∞ ✈➻ |x − y|−s ≥ ri−s tr➯♥ Ai ◆❤÷♥❣✱ Is (à) = s (x)dà(x) < ợ s (x) = dà(y) |xy|s õ s (x) < ợ µ✲❤➛✉ ❦❤➢♣ ♥ì✐✳ ❱➟②✱ µ(F1 ) = ❱➻ lim µ(B(x,r)) = ♥➳✉ x ∈ (F \ F1 ) ♥➯♥ tø ▼➺♥❤ ✤➲ ✷✳✷✳✺ rs t❛ ❝â ∀c > t❤➻ r→0 Hs (F ) ≥ Hs (F \ F1 ) ≥ µ(F \ F1 ) µ(F ) − µ(F1 ) µ(F ) ≥ = c c c ❉♦ ✤â✱ Hs (F ) = ∞ ✈➔ ❞✐♠H F ≥ s ✐✐✮ ●✐↔ sû Hs (F ) > ❚❛ ❞ị♥❣ Hs ✤➸ ①➙② ❞ư♥❣ ♠ët ♣❤➙♥ ❜è ❦❤è✐ ữủ tr F ợ It (à) < , t < s tỗ t t ❝♦♠♣❛❝t E ⊂ F ✈ỵ✐ < Hs (E) < ∞ ✈➔ ♠ët ❤➡♥❣ sè b ✤➸ Hs (E ∩ B(x, r)) ≤ brs ✈ỵ✐ ∀x ∈ Rn , r > ▲➜② µ ❧➔ t❤✉ ❤➭♣ ❝õ❛ Hs ❧➯♥ E ✱ ♥❣❤➽❛ ❧➔ µ(A) = Hs (E∩A), ✷✶ ❦❤✐ õ sỹ ố ố ữủ tr F ❈è ✤à♥❤ x ∈ Rn ✈➔ ✤➦t m(r) = µ(B(x, r)) = Hs (E ∩ B(x, r)) ≤ brs ❑❤✐ ✤â✱ ♥➳✉ < t < s t❛ ❝â dµ(x) + t |x−y|≤1 |x − y| φt (x) = ≤ dµ(y) t |x−y|>1 |x − y| r−t dm(r) + µ(Rn ) = [r−t m(r)] |10 +t r−t+1 m(r)dr + µ(Rn ) ≤ b + bt rs−t−1 dr + µ(Rn ) = b(1 + t s s−t ) + H (F ) = c ◆❤÷ ✈➟②✱ φt (x) ≤ c, ∀x ∈ Rn , s❛♦ ❝❤♦ Is (µ) = φt dµ(x) ≤ cµ(Rn ) < ∞ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✸ ✣➦❝ tr÷♥❣ t tỹ ỗ õ sr ữỡ ởt số t tỹ ỗ ■❋❙ t❤ä❛ ♠➣♥ ❖❙❈ ❛✮ ❚➟♣ ❈❛♥t♦r✳ ●å✐ C ❧➔ t➟♣ ❈❛♥t♦r ✭✈ỵ✐ t➟♣ ❈❛♥t♦r ♠ỉ t↔ ð ♠ư❝ ✷✳✶✳✶ C s ỗ ỗ {f1 , f2 } ợ f : [0, 1] → [0, 1] f1 : [0, 1] → [0, 1] ✈➔ x → 13 x + 32 x → 3x ❑❤✐ ✤â ❧➜② t➟♣ ♠ð ❱ ❂ ✭✵✱✶✮ = ∅ t❤➻ f1 (V ) = (0, 13 )✱ f2 (V ) = ( 23 , 1) ❞➝♥ ✤➳♥ f1 (V ) ∩ f2 (V ) = φ f1 (V ) ∪ f2 (V ) ⊂ V ❉♦ ✤â {f1 , f2 } ❧➔ ■❋❙ t❤ä❛ ♠➣♥ ❖❙❈✳ ❜✮ ❇ư✐ ❝❛♥t♦r✳ ●å✐ B ❧➔ ❇ư✐ ❈❛♥t♦r ✭✈ỵ✐ ❇ư✐ ❈❛♥t♦r ♠ỉ t↔ ð ♠ư❝ ✷✳✶✳✶ ✭❜✮✮✱ B s✐♥❤ ỗ ỗ {f1 , f2 , f3 , f4 } ✈ỵ✐ fi : [0; 1] × [0; 1]→[0; 1] × [0; 1], i = 1, 2, 3, ①→❝ ✤à♥❤ ❜ð✐ f1 (x, y) = x y + , 4 , f2 (x, y) = ✷✷ x y + , + 4 4 , f3 (x, y) = x y + , + 4 , f4 (x, y) = x y , + 4 ❑❤✐ ✤â✱ ❧➜② V = (0, 1) × (0, 1) t❤➻ t❛ ❝â fi (V ) ∩ fj (V ) = φ, ∀i = j f1 (V ) ∪ f2 (V ) ∪ f3 (V ) ∪ f4 (V ) ⊂ V ❉♦ ✤â {f1 , f2 , f3 , f4 } ❧➔ ■❋❙ t❤ä❛ ♠➣♥ ❖❙❈ ✳ ✣➸ ❝❤➾ r❛ ✤➦❝ trữ t t tỹ ỗ õ s ữỡ trữợ t t s ✷✳✸✳✷ ❇ê ✤➲ ([3])✳ ●✐↔ sû {Vi} ❧➔ ❞➣② ❝→❝ t➟♣ ♠ð rí✐ ♥❤❛✉ tr♦♥❣ Rn s❛♦ ❝❤♦ ♠é✐ t➟♣ Vi ✤➲✉ ❝❤ù❛ ♠ët ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ a1 r ✈➔ Vi ❧↕✐ ❝❤ù❛ ♠ët ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ a2 r✳ ❑❤✐ ❞â✱ ♠å✐ ❤➻♥❤ ❝➛✉ B ❜→♥ ❦➼♥❤ r ❝â ❣✐❛♦ ❦❤→❝ ré♥❣ ✈ỵ✐ ♥❤✐➲✉ ♥❤➜t (1 + 2a2 )n a−n ❝→❝ ❜❛♦ ✤â♥❣ Vi ❝õ❛ Vi ✷✳✸✳✸ ([3]) ỵ sỹ ố ố ❧÷đ♥❣✮✳ ❈❤♦ ❋ ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t✱ ❦❤→❝ ré♥❣ tr♦♥❣ Rn ởt sỹ ố ố ữủ tr ợ ộ s tỗ t c > ✈➔ δ > s❛♦ ❝❤♦ µ(U ) ≤ c|U |s ✈ỵ✐ ♠å✐ ❯ ♠➔ |U | ≤ δ t❤➻ Hs (F ) ≥ µ(F ) c > ❉♦ ✤â dimH F ≥ s ❚ø ❇ê ✤➲ ✷✳✸✳✸ t❛ ❝â ✤à♥❤ ❧➼ s❛✉ ✈➲ ✤➦❝ tr÷♥❣ ❝õ❛ t➟♣ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣ ✷✳✸✳✹ ✣à♥❤ ❧➼ ([3]) ỗ ỗ ợ t số ỗ tữỡ ự ci ∈ (0; 1), i ∈ {1, 2, , m} tr➯♥ Rn t❤ä❛ ♠➣♥ ❖❙❈✱ ✈➔ ❋ ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ❤➺ ❤➔♠ ❧➦♣✳ ❑❤✐ ✤â✱ dimH (F ) = s ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ m i=1 csi ❂ ✶✳ ❍ì♥ ♥ú❛✱ t❛ ❝â Hs (F ) > ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❧➔ s ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ m i=1 csi ❂ ✶✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❞✐♠H (F ) = s t❛ ❝❤ù♥❣ ♠✐♥❤ ❞✐♠H (F ) s H (F ) s rữợ ❤➳t✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❞✐♠H (F ) ≤ s✳ ❱ỵ✐ ♠å✐ t➟♣ ❆ ✈➔ ♠å✐ ❜ë sè ❦ tü ♥❤✐➯♥ ij ∈ {1, 2, , m}, j = (1, 2, , k) t❛ ❦➼ ❤✐➺✉ Ai1 ,i2 , ,ik = Si1 ◦ Si2 ◦ ◦ Sik (A)✳ ✷✸ ❉➵ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ Si1 ◦ Si2 ◦ ◦ Sik ỗ ợ t số ỗ ❧➔ ci1 ci2 cim ✳ ✣➦t Jk = {(i1 , i2 , , ik )} : ij ∈ {1, 2, , m}, j = (1, 2, , k) ❚❛ ❝â m m m k Sik (F ) ) ( ( F = S (F ) = ik i2 i1 = Fi1 ,i2 , ,ik (i1 ,i2 , ,ik )∈Jk ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ δ > t❛ ❝â t❤➸ ❝❤å♥ ❦ ✤õ ❧ỵ♥ s❛♦ ❝❤♦ |Fi1 ,i2 , ,ik | ≤ (maxci )k |F | ≤ δ ✭✈➻ |Fi1 ,i2 , ,ik | = |Si1 ◦ Si2 ◦ ◦ Sik (F )| = ci1 ci2 cik |F |.✮ ❱➻ ✈➟②✱ ❤å {Fi1 ,i2 , ,ik } ✈ỵ✐ (i1 , i2 , , ik ) ∈ Jk ❧➔ ♠ët δ− ♣❤õ ❋✳ ❚❛ ❝â m s s (ci1 ci2 cik ) |F | = ik m csi2 i=1 ✭✈➻ s ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ m ♥➔② ❞➝♥ ✤➳♥ ≤ |F |s csik i=1 ij =1 Hs (F ) m csi1 |F |s = |F |s , i=1 csij ❂ ✶ ✮✳ ❉♦ ✤â ✱ Hδs (F ) ≤ |F |s ✣✐➲✉ < +∞✱ ❤❛② t❛ ❝â dimH (F ) ≤ s✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❧➔ dimH (F ) ≥ s ỵ I = {(i1 , i2 , ) : ≤ ij ≤ m} ✈➔ ✈ỵ✐ ❞➣② {(i1 , i2 , , ik )} trữợ ỵ Ii1 ,i2 , ,ik = {(i1 , i2 , , ik , qk+1 , ) : qj m, j > k} ỗ ỳ tr♦♥❣ I ✈ỵ✐ k sè ❤↕♥❣ ✤➛✉ ❧➔ i1 , i2 , , ik ✳ ❚❛ ①➙② ỹ ởt sỹ ố ố ữủ tr I ❜ð✐ µ(Ii1 ,i2 , ,ik ) = (ci1 cik )s ✳ ❚ø m s m s csi (ci1 cik ) = (ci1 cik ) i=1 ♥❣❤➽❛ ❧➔ (ci1 cik ci )s , = i=1 m µ(Ii1 ,i2 , ,ik ) = µ(Ii1 ,i2 , ,ik ,i ), i=1 ✷✹ ❦❤✐ ✤â µ ❧➔ ♠ët sü ♣❤➙♥ ❜è ❦❤è✐ ữủ tr t I ợ à(I) = 1✳ ❚ø ✤â t❛ t❤✐➳t ❧➟♣ ♠ët sü ♣❤➙♥ ❜è ố ữủ tr F ữ s ợ ộ t A ∞ ❝õ❛ F ✱ ✤➦t µ(A) = µ{(i1 , i2 , ) : xi1 ,i2 , = Fi ∈ A}✳ ❚❛ ❝❤➾ r❛ r➡♥❣ µ tọ i=1 ỵ sỹ ố ố ❧÷đ♥❣ tr➯♥ F ✳ ❚❤➟t ✈➟②✱ ❞➵ ❦✐➸♠ tr❛ ✤÷đ❝ µ(F ) = 1✳ ●✐↔ sû V ❧➔ t➟♣ ♠ð t❤ä❛ ♠➣♥ ✭✶✳✶✮✳ ❉♦ V ⊃ S(V ) = m Si (V )✱ ❞➣② i=1 ❣✐↔♠ ❝→❝ t➟♣ S k (V ) ❤ë✐ tö ✈➲ F ✳ ✣➦❝ ❜✐➺t V ⊃ F ✈➔ V i1 ,i2 , ,ik ⊃ Fi1 ,i2 , ,ik ✈ỵ✐ ♠é✐ ❞➣② ❤ú✉ ❤↕♥ (i1 , i2 , , ik )✳ ●å✐ B ❧➔ ❤➻♥❤ ❝➛✉ ❜➜t ❦ý ❜→♥ ❦➼♥❤ r < 1✳ ữợ ữủ à(B) t t Vi1 ,i2 , ,ik ❝â ✤÷í♥❣ ❦➼♥❤ ❝â t❤➸ s♦ s→♥❤ ữủ ợ ữớ B F B ✳ ❚✐➳♣ t❤❡♦✱ t❛ rót ❣å♥ ♠é✐ ❞➣② ✈ỉ ❤↕♥ (i1 , i2 , ) ∈ I tø ❝❤➾ sè t❤ù k s❛♦ ❝❤♦ ci r ≤ ci1 cik ≤ r 1≤i≤m Q t ỗ tt ỳ ❞➣② ✭❤ú✉ ❤↕♥✮ ✤↕t ✤÷đ❝ t❤❡♦ ❝→❝❤ t❤✉ ❣å♥ tr➯♥✳ ❑❤✐ ✤â✱ ♠é✐ ❞➣② ✈æ ❤↕♥ (i1 , i2 , ) ∈ I ✤➲✉ ❝â ✤ó♥❣ ♠ët ❣✐→ trà k s❛♦ ❝❤♦ (i1 , i2 , , ik ) t❤✉ë❝ Q✳ ❉♦ V1 , , Vm ❦❤ỉ♥❣ ❣✐❛♦ ♥❤❛✉ ♥➯♥ ✈ỵ✐ ♠é✐ (i1 , i2 , , ik ) ∈ Q t❤➻ ❝→❝ ♣❤➛♥ tû ❝õ❛ ❞➣② Vi1 ,i2 , ,ik ,1 , , Vi1 ,i2 , ,ik, m ❝ơ♥❣ ❦❤ỉ♥❣ ❣✐❛♦ ♥❤❛✉✳ ❉♦ ✤â✱ ❧ỵ♣ ❝→❝ t➟♣ ♠ð {Vi1 ,i2 , ,ik : (i1 , i2 , , ik ) ∈ Q} ❧➔ ❦❤ỉ♥❣ ❣✐❛♦ ♥❤❛✉✳ ❍ì♥ ♥ú❛✱ t❛ ❝â F ⊂ Fi1 , ,ik ⊂ Q V i1 , ,ik Q ❈❤å♥ a1 ✈➔ a2 s❛♦ ❝❤♦ V ❝❤ù❛ ởt a1 ỗ tớ V ❧↕✐ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ a2 ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ (i1 , i2 , , ik ) ∈ Q t❤➻ t➟♣ Vi1 ,i2 , ,ik ❝❤ù❛ ♠ët ❤➻♥❤ ❝➛✉ ci1 cik a1 ✈➔ Vi1 ,i2 , ,ik ❧↕✐ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ ci1 cik a2 ✳ ❉♦ ✤â✱ ♥❤í ✭✷✳✺✮ t❛ ❝â Vi1 ,i2 , ,ik ❝❤ù❛ ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ ( ci )a1 r ✈➔ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ a2 r✳ ▲➜② Q1 ❧➔ 1≤i≤m ♥❤ú♥❣ ❞➣② (i1 , i2 , , ik ) ∈ Q s❛♦ ❝❤♦ B ❝â ❣✐❛♦ ✈ỵ✐ V i1 ,i2 , ,ik ✳ ❚❤❡♦ ❇ê ✤➲ ✷✺ −n ❞➣② tr♦♥❣ Q ✳ ❑❤✐ ✤â✱ ✷✳✸✳✷ ❝â ♥❤✐➲✉ ♥❤➜t q = (1 + 2a2 )n a−n 1 (mini ci ) µ(B) = µ(F ∩ B) = µ{(i1 , i2 , ) : xi1 ,i2 ,i3 ∈ F ∩ B} ≤ µ( Ii1 ,i2 , ,ik ) Q1 ◆➯♥✱ ♥➳✉ xi1 ,i2 , ∈ F ∩ B ⊂ V i1 ,i2 , ,ik t❤➻ ❝â ✤ó♥❣ ♠ët sè tü ♥❤✐➯♥ k s❛♦ Q1 ❝❤♦ (i1 , i2 , , ik ) ∈ Q1 ✳ ❱➻ ✈➟②   Ii1 ,i2 , ,ik  ≤ µ(B) ≤ µ  Q1 Q1 (ci1 cik )s ≤ = Q1 ❉♦ ✤â✱ ❜➜t ❦ý ♠ët t➟♣ U ⊂ µ(Ii1 ,i2 , ,ik ) rs = qrs (1.2) Q1 Rn ✤➲✉ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ♠ët ❤➻♥❤ ❝➛✉ ❝â ✤÷í♥❣ ❦➼♥❤ ❧➔ |U |✳ ▼➔ t❤❡♦ ✭✷✳✺✮ t❛ ❝â µ(U ) ≤ q|U |s ✳❉♦ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✸✳✸ t❛ ❝â Hs (F ) ≥ q −1 > 0✳ ❱➻ ✈➟② t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❞✐♠H F t❛ ❝â ❞✐♠H F ≥ s✳ ❱➟② t❛ ❝â ❞✐♠H F = s✳ ◆❤➟♥ ①➨t✳ ◆➳✉ F s✐♥❤ ❜ð✐ ■❋❙ t❤ä❛ ♠➣♥ ❖❙❈ ✈ỵ✐ s = ❞✐♠H F t❤➻ Hs (F ) > ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ❦➳t q✉↔ ✈➲ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❝õ❛ ❦➳t q✉↔ tr➯♥✳ ▼ët sè ❦➼ ❤✐➺✉ ✣➦t A = {1, 2, , m}✱ A∗ = {An : n = 0, 1, , }✱ s ∈ A tỗ t p N s = s1 s2 sp , si ∈ A, i = 1, , p✳ ❚❛ ❦➼ ❤✐➺✉ ✤ë ❞➔✐ ❝õ❛ s ❧➔ |s| = p✳ ●✐↔ sû s = s1 s2 sp , t = t1 t2 tq ❚❛ ✈✐➳t s ⊆ t ♥➳✉ p ≤ q ✈➔ sk = tk ✈ỵ✐ k = 1, 2, , p✳ ❚❛ ♥â✐ s ✈➔ t ❧➔ ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝ ♥➳✉ ❦❤ỉ♥❣ ❝â s ⊆ t ✈➔ t ⊆ s ♥❣❤➽❛ ❧➔ sk = tk ✈ỵ✐ k ♥➔♦ ✤â ❜➨ ❤ì♥ ❤♦➦❝ ❜➡♥❣ min{p, q}✳ ✣➦t rs = rs1 , rs2 rsp , Ss = Ss1 , Ss2 Ssp ✈➔ As = Ss (A)✳ ❑➼ ❤✐➺✉ F = {Ss−1 St : s, t ∈ A∗ ; s, t ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝ } ✭✷✳✻✮ ✷✳✸✳✺ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✸❪✮ ❈❤♦ G ❧➔ t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣ ✈➔ ♠ët ♣❤➨♣ t♦→♥ ✷ ✷✻ ♥❣æ✐ (x, y) → xy tø G × G → G ❑❤✐ ✤â✱ (G, ) ❧➔ ♠ët ♥❤â♠ ♥➳✉ ♣❤➨♣ t♦→♥ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✮ P❤➨♣ t♦→♥ ❧➔ ❦➳t ❤ñ♣✱ tù❝ ❧➔ a(bc) = (ab)c ✈ỵ✐ ♠å✐ a, b ✈➔ c ∈ G ỗ t e G s ea = ae = a ✈ỵ✐ ♠å✐ a ∈ G❀ ✸✮ ợ a G tỗ t b G s❛♦ ❝❤♦ ab = ba = e ❑❤✐ ✤â✱ e ❧➔ ❞✉② ♥❤➜t ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ ✈à ❝õ❛ ♥❤â♠ G ✷✳✸✳✻ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✸❪✮ ❈❤♦ X = ∅✱ ❤å τ ❝→❝ t➟♣ ❤đ♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ♠æt tæ♣æ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✿ ✶✮ ∅, X ∈ τ ; ✷✮ ◆➳✉ U1 , U2 ∈ τ t❤➻ U1 ∩ U2 ∈ τ ; ✸✮ ❱ỵ✐ ♠å✐ ❤å {Uα }α ∈ I ⊂ τ ❑❤✐ ✤â✱ (X, τ ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✳ ✷✳✸✳✼ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✸❪✮ ▼ët ♥❤â♠ tæ♣æ ❧➔ ♠ët ♥❤â♠ G ✤÷đ❝ tr❛♥❣ ❜à tỉ♣ỉ τ s❛♦ ❝❤♦ ♣❤➨♣ t♦→♥ (x, y) → xy ✈➔ →♥❤ ①↕ x → x−1 tø G × G → G ❧➔ ❧✐➯♥ tö❝✳ ✷✳✸✳✽ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ●✐↔ sû ● ❧➔ ♠ët ♥❤â♠ ✈➔ τ ❧➔ ♠ët tæ♣æ tr➯♥ G✳ ❑❤✐ ✤â✱ (G, τ ) ❧➔ ♠ët ♥❤â♠ tæ♣æ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ →♥❤ ①↕ (a, b) → ab−1 ❧➔ ❧✐➯♥ tư❝ tr➯♥ G × G ❚❛ ❝â t❤➸ ①❡♠ F ✤÷đ❝ ♠ỉ t↔ ð ✭✷✳✻✮ ♥❤÷ ❧➔ t➟♣ ❝♦♥ ❝õ❛ õ tổổ G ỗ tr D ỵ s ợ t số ữỡ (r, s) t ổ tỗ t↕✐ ➼t ♥❤➜t ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ R(r, s) s❛♦ ợ t ỗ t tr ✤➾♥❤ ❧➔ R(r, s) t❤➻ ❝→❝ ❣â❝ ❝õ❛ ♥â ❧➔ ọ t ổ tỗ t ởt ỗ t tr r ợ ỗ t ởt ỗ t tr s ợ ỗ t t ọ ỵ sỷ f1, f2, , fm ỗ ❑➼ ❤✐➺✉ ❆ ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ❤➺ ❤➔♠ ❧➦♣ ✤â ✈➔ ❋ ❧➔ t➟♣ ❝→❝ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ✷✼ ❜ð✐ ✭✷✳✻✮ ❦➳t ❤đ♣ ✈ỵ✐ ❤➺ {f1 , f2 , , fm }✳ ❑❤✐ ✤â✱ t❛ ❝â Hα (A) > ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ id ∈ / cl(F ) ✈ ỵ✐ α ❧➔ ❝❤✐➲✉ s ỗ t õ tr ự rữợ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ●✐↔ sû id ∈ / cl(F ) t❛ ❝❤ù♥❣ ♠✐♥❤ µα (A) > 0✳ ❚❛ ❝â t❤➸ ❣✐↔ sû A ❦❤ỉ♥❣ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ♠ët s✐➯✉ ♣❤➥♥❣ ♥➔♦ tr♦♥❣ Rn ✳ ❈❤å♥ x0 , x1 , , xd ❧➔ (d + 1) ✤✐➸♠ tr♦♥❣ A✳ ❑❤✐ ✤â✱ ∃ε > s❛♦ ❝❤♦ Wε ∩ F = φ s❛♦ ❝❤♦ Wε ∩ F = φ ✈ỵ✐ Wε = {h ∈ G : |h(xj ) − xj | < ε, j = 0, 1, , d, h = fs−1 ft } ✣✐➲✉ ♥➔② ♥❣❤➽❛ ❧➔ ✈ỵ✐ s, t ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝✱ ∃j ∈ {0, 1, , d} ♠➔ |fs−1 ft (xj ) − xj | ≥ ε ❞➝♥ ✤➳♥ |ft (xj ) − fs (xj )| ≥ rs ε ❑➼ ❤✐➺✉ α = |A| = ❞✐❛♠A t❤➻ |As | = rs a✳ ❇➙② ❣✐í t❛ r r tỗ t số k s ✈ỵ✐ ❜➜t ❦ý t➟♣ U ❤å s❛✉ ❝â ➼t ♥❤➜t k ♣❤➛♥ tû✿ C = {s = s1 sp ∈ S ∗ : rs a < ❞✐❛♠U ≤ rs1 rsp−1 a, As ∩ U = φ} ◆➳✉ s, t ∈ C ❧➔ ♣❤➙♥ ❜✐➺t t❤➻ ❝❤ó♥❣ ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝ ✈➔ |ft (xj ) − fs (xj )| ≥ rs ε ≥ rε |U | a ✈ỵ✐ j = j(s, t) ∈ {0, 1, , d} ✈➔ r = min{r1 , , rm }✳ ❇➙② ❣✐í ❣✐↔ sû r➡♥❣ C ❧➔ ❤å ❝♦♥ ❝õ❛ C s❛♦ ❝❤♦ j(s, t) ✈ỵ✐ ∀s, t ∈ C ❑❤✐ ✤â✱ ✈ỵ✐ ρ = rε 2a t❤➻ ❤➻♥❤ ❝➛✉ t➙♠ fs (xj ) ❜→♥ ❦➼♥❤ ρ.|U |✱ s ∈ C ❧➔ rí✐ ♥❤❛✉ ✈➔ t➙♠ ❝õ❛ ♥â fs (xj ) ∈ As ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ♠ët ❤➻♥❤ ❝➛✉ ❝è ✤à♥❤ ❜→♥ ❦➼♥❤ 2|U | t➙♠ tr♦♥❣ U ✳ ❙û ❞ö♥❣ ✤ë ✤♦ ▲❡❜❡s❣✉❡ t❛ t❤➜② r➡♥❣ sè ❝→❝ ❤➻♥❤ ❝➛✉ ❧➔ ❈❛r❞ C < (2ρ)d /ρd := N ỵ s t ợ t N d tỗ t số K = K(N, d) s❛♦ ❝❤♦ ❈❛r❞ C ≤ k ✳ ❱ỵ✐ ✤ë ✤♦ tü ♥❤✐➯♥ µ tr➯♥ A ✈➔ t➟♣ ❤đ♣ ❜➜t ❦ý t❛ ❝â µ(U ) ≤ {µ(As ) : s ∈ C} = U {rsα : rsα ∈ C} < k.| |α a ◆❤÷ ✈➟②✱ ❜➜t ❦ý ♣❤õ U1 , U2 , , ❝õ❛ A t❛ ❝â ∞ |Uj |α ≥ j=1 ✈➻ µ(A) = ❉♦ ✤â✱ α µα (A) ak aα k > α ( ak ) ∞ µ(Uj ) ≥ j=1 aα k > ❈❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ trữợ t t ự s ❑❤➥♥❣ ✤à♥❤✳ ◆➳✉ f1 , , fm ỗ ợ t số ỗ ri ỗ t ❜➜t ❜✐➳♥ q✉❛ ❤➺ ❤➔♠ ❧➦♣ ♥➔② ♥❣❤➾❛ ❧➔ α ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✿ m i=1 riα ❂ ✶✳ ❑❤✐ ✤â ✐✮ ❱ỵ✐ B ⊂ A ❧➔ t➟♣ ✤♦ ữủ t (B) = (B) = inf{ |Ui |α : Ui ♠ð, i∈I Ui ⊃ B} i∈I ✐✐✮ µα (fs (A) ∩ ft (A)) = ✈ỵ✐ s, t ∈ S ∗ ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝✳ ❚❤➟t ✈➟②✱ ố ợ ự trữợ t t t B = A✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ t❛ ❝â µ∗ ≤ µα ❚❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ✈ỵ✐ ♠é✐ ♣❤õ ♠ð {Ui } ❝õ❛ A ộ > ổ tỗ t ❧➔ {Vj }✱ |Vj | < δ ✈ỵ✐ ∀j ✈➔ m |Vi |α = |Ui |α ❈❤å♥ n s❛♦ ❝❤♦ rs < δ| Ui | ✈ỵ✐ s ∈ S n ✈➔ t❛ t❤❛② ♠é✐ i=1 Ui ❜ð✐ ❤å {fs (Ui ) : s ∈ S n }✳ ❚➟♣ ♥➔② ♣❤õfs (A)✱ ❞♦ ✤â ♥â ♣❤õ A ✈➔ tê♥❣ ❝á♥ ❧↕✐ ổ t t tỹ ỗ ợ ộ B A t õ (A) = µα (B) + µα (A \ B) ≥ µ∗ (B) + µ∗ (A \ B) ≥ µ∗ (A) = µα (A), ❦➨♦ t❤❡♦ µ∗ (B) = µα (B)✳ ❚ø µ∗ ( i∈S fi (A)) = µα (A) = i∈S riα µ(A) = µα (f (A)) t❛ ❦➳t ❧✉➟♥ r➡♥❣ i∈S µα (fi (A) ∩ fj (A)) = ✈ỵ✐ i = j tữỡ tỹ tờ ợ ♠å✐ s ∈ S n ✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝õ❛ ✤à♥❤ ❧➼✳ ●✐↔ sû r➡♥❣ id ∈ cl(F ) ✈➔ µα (A) > 0✳ ▲➜② η ∈ (1; 32 ) ✈➔ ♠å✐ ♣❤õ ♠ð {U1 , U2 , , Un } ❝õ❛ A ợ |Ui | < (A) U = m Ui ✈➔ δ = inf {|a − x| : a ∈ A, x ∈ U }✳ ❱➻ i=1 id ∈ cl(F ) ♥➯♥ ∃s, t ∈ S ∗ ❦❤æ♥❣ s s ữủ ợ (rs1 rt ) η ✭✷✳✼✮ ✈➔ sup{|x − fs−1 |.ft (x) : x ∈ A} ❁δ ✳ ◆❤÷ ✈➟② fs−1 ft (A) ≤ U ✈➔ ft (A) ∪ fs (U )✳ ❚❤❡♦ ❦➳t q✉↔ ✭✐✮ ✈➔ ✭✐✐✮ ✈ø❛ ❝❤ù♥❣ ♠✐♥❤ t❛ ❝â µα (ft (A)) + µα (fs (A)) = µα (ft (A) ∪ fs (A)) n ≤ i=1 = rsα |fs (Ui |α |Ui |α < rsα η.µα (A)✳ α , ❦➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ✭✷✳✼✮ t❛ ❝â − η ≤ η − ▼➙✉ ❉♦ ✤â✱ rαt ≤ (à 1)r t ợ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✸✵ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉ ✶✳ ❚➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ✈➲ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤ë ✤♦ ❍❛✉s❞♦❢❢✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣ ð ✣à♥❤ ❧➼ ✷✳✶✳✶✱ ✷✳✷✳✷✱ ✷✳✷✳✹✱ ✷✳✷✳✺✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ✈➲ ✤✐➲✉ ❦✐➺♥ ✤➸ ♠ët t➟♣ ❝â ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❞÷ì♥❣ t❤➸ ❤✐➺♥ tr♦♥❣ ❈❤÷ì♥❣ ✷✱ ð ❝→❝ ✣à♥❤ ❧➼ ✷✳✸✳✸✱ ✷✳✸✳✹✱ ✷✳✹✳✶✱ ✷✳✹✳✷✳ ✸✳ ❚➻♠ ✤÷đ❝ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ✈➲ t➟♣ ❋r❛❝t❛❧ ❝â ✤ë ✤♦ ❍❛✉s❞♦❢❢ ❞÷ì♥❣ ✭❱➼ ❞ư ✷✳✶✳✶ ❛✱❜✮✳ ✸✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❈✳❇❛♥❞t ❛♥❞ ❙✳●r❛❢ ✭✶✾✾✷✮✱ ❙❡❧❢✲s✐♠✐❧❛rs❡ts✼✳ ❆❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❙❡❧❢✲ s✐♠✐❧❛r ❢r❛❝t❛❧ ✇✐t❤ ♣♦s✐t✐✈❡ ❤❛✉s❞♦❢❢ ♠❡❛s✉r❡✱ ♣r♦❝✳ ❆♠❡r✐❝❛♥ ▼❛t❤❡✲ ✶✶✹ ✭✹✮✱ ✾✾✺✲✶✵✵✶✳ ♠❛t✐❝❛❧ s♦❝✐❡t②✱ ❱♦❧ ❬✷❪ ●✳ ❊❞❣❛r ✭✷✵✵✼✮✱ ▼❡❛s✉r❡✱ ❚♦♣♦❧♦❣②✱ ❛♥❞ ❋r❛❝t❛❧ ●❡♦♠❡tr②✱ ❙♣r✐♥❣❡r✳ ❬✸❪ ❑✳ ❋❛❧❝♦♥❡r ✭✶✾✾✵✮✱ ❋r❛❝t❛❧ ●❡♦♠❡tr②✱ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦✉♥❞❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❏♦❤♥ ❲✐❧❡②✳ ❬✹❪ ❨✳ P❡r❡s✱ ▼✳❘❛♠s✱ ❑✳❙✐♠♦♥✱ ❛♥❞ ❇✳❙♦❧♦♠♦❢❛❦ ✭✷✵✵✶✮✱ ❊q✉✐✈❛❧❡♥❝❡ ♦❢ ♣♦s✲ ✐t✐✈❡ ❤❛✉s❞♦❢❢ ♠❡❛s✉r❡ ❛♥❞ t❤❡ ♦♣❡♥s❡t ❝♦♥❞✐t✐♦♥ ❢♦r ❙❡❧❢✲❝♦♥❢♦r♠❛❧ s❡ts✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧s♦❝✐❡t②✱ ❱♦❧ ✶✷✾ ✭✾✱ ✷✻✽✾✲✷✻✾✾✮ ❏✳ ❊✳ ❍✉t❝❤✐♥s♦♥ ✭✶✾✽✶✮✱ ❋r❛❝t❛❧ ❛♥❞ ❙❡❧❢✲s✐♠✐r❛❧✐t②✱ ■❞✐❛♥❛ ❯♥✐✈✳ ▼❛t❤✳ ❏✱ ❱♦❧ ✸✵✱ ✼✶✸✲✼✹✼ ❬✺❪ ◆✳ P❛t③s❝❤❦❡ ✭✶✾✾✼✮✱ s❡❧❢✲❝♦♥❢♦r♠❛❧ ♠✉❧t② ❢r❛❝t❛❧ ♠❡❛s✉r❡s✱ ❆❞✈✳ ❆♣♣❧✳ ✶✾✱ ✹✽✻✲✺✶✸ ▼❛t❤✱ ❱♦❧✳ ✸✷