❍❆◆❖■ P❊❉❆●❖●■❈❆▲ ❯◆■❱❊❘❙■❚❨ ✷ ❉❊P❆❘❚▼❊◆❚ ❖❋ ▼❆❚❍❊▼❆❚■❈❙ ▲❯❖◆● ❚❍■ ❚❯❨❊◆ ❈❖❱❊❘■◆● ❚❍❊❖❘❊▼❙ ❆◆❉ ❆PP▲■❈❆❚■❖◆❙ ●❘❆❉❯❆❚■❖◆ ❚❍❊❙■❙ ❍❛♥♦✐✱ ✷✵✶✾ ❍❆◆❖■ P❊❉❆●❖●■❈❆▲ ❯◆■❱❊❘❙■❚❨ ✷ ❉❊P❆❘❚▼❊◆❚ ❖❋ ▼❆❚❍❊▼❆❚■❈❙ ▲❯❖◆● ❚❍■ ❚❯❨❊◆ ❈❖❱❊❘■◆● ❚❍❊❖❘❊▼❙ ❆◆❉ ❆PP▲■❈❆❚■❖◆❙ ❙♣❡❝✐❛❧✐t②✿ ❆♥❛❧②s✐s ●❘❆❉❯❆❚■❖◆ ❚❍❊❙■❙ ❙✉♣❡r✈✐s♦r✿ ❉r✳ ❉♦ ❍♦❛♥❣ ❙♦♥ ❍❛♥♦✐✱ ✷✵✶✾ ❈♦♥❢✐r♠❛t✐♦♥ ■ ❛ss✉r❡ t❤❛t t❤❡ r❡s✉❧ts ✐♥ t❤✐s t❤❡s✐s ❛r❡ tr✉❡ ❛♥❞ t❤❡ t♦♣✐❝ ♦❢ t❤✐s t❤❡s✐s ✐s ♥♦t ✐❞❡♥t✐❝❛❧ t♦ ♦t❤❡r t♦♣✐❝✳ ■ ❛❧s♦ ❛ss✉r❡ t❤❛t ❛❧❧ t❤❡ ❤❡❧♣ ❢♦r t❤✐s t❤❡s✐s ❤❛s ❜❡❡♥ ❛❝❦♥♦✇❧✲ ❡❞❣❡ ❛♥❞ t❤❛t t❤❡ ✉s❡❞ ❧✐t❡r❛t✉r❡ ❛♥❞ ♦t❤❡r ❛✉①✐❧✐❛r② r❡s♦✉r❝❡s ❤❛✈❡ ❜❡❡♥ ❝♦♠♣❧❡t❡❧② r❡❢❡r❡♥❝❡❞✳ ❚❤❡ ❛✉t❤♦r ▲✉♦♥❣ ❚❤✐ ❚✉②❡♥ ❆❝❦♥♦✇❧❡❞❣♠❡♥t ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② ❞❡❡♣ ❣r❛t✐t✉❞❡ t♦ ♠② s✉♣❡r✈✐s♦r✱ ❉r✳ ❉♦ ❍♦❛♥❣ ❙♦♥✱ ❢♦r ❤✐s ❝❛r❡❢✉❧ ❛♥❞ ❡❢❢❡❝t✐✈❡ ❣✉✐❞❛♥❝❡✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ t❤❡ ❜♦❛r❞ ♦❢ ❞✐r❡❝t♦rs ♦❢ ❍❛♥♦✐ P❡❞❛❣♦❣✐❝❛❧ ❯♥✐✈❡rs✐t② ✷✱ ❢♦r ♣r♦✈✐❞✐♥❣ ♠❡ ✇✐t❤ ♣❧❡❛s❛♥t ✇♦r❦✐♥❣ ❝♦♥❞✐t✐♦♥s✳ ■ ❛♠ ❣r❛t❡❢✉❧ t♦ t❤❡ ❧❡❛❞❡rs ♦❢ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❛♥❞ ♠② ❝♦❧❧❡❛❣✉❡s✱ ❢♦r ❣r❛♥t✐♥❣ ♠❡ ✈❛r✐♦✉s ❢✐♥❛♥❝✐❛❧ s✉♣♣♦rts ❛♥❞✴♦r ❝♦♥st❛♥t ❤❡❧♣ ♠② st✉❞②✳ ❈♦♥t❡♥ts ■♥tr♦❞✉❝t✐♦♥ ✶ ✶ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ✷ ✶✳✶✳ ❆ ✺r✲❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ✷ ✶✳✷✳ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ❘❛❞♦♥ ♠❡❛s✉r❡s ✷✳✶✳ ✷✳✷✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇❡s✐❝♦✈✐t❝❤✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ✹ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ✶✺ ✐ ❚❛❜❡❧ ♦❢ ♥♦t❛t✐♦♥ ❲❡ ✐♥tr♦❞✉❝❡ ❤❡r❡ t❤❡ ♥♦t❛t✐♦♥ ❢♦r s♦♠❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ✇❤✐❝❤ ❛r❡ ♥♦t ❞❡❢✐♥❡❞ ✐♥ t❤❡ t❤❡s✐s✳ Z✱ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✳ R✱ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ Rn , t❤❡ n✲ ❞✐♠❡♥s✐♦♥❛❧ ❡✉❝❧✐❞❡❛♥ s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ✐♥♥❡r ❛♥❞ t❤❡ ♥♦r♠ |x|✳ n−1 S = {x ∈ Rn : |x| = 1} , t❤❡ ✉♥✐t s♣❤❡r❡✳ [a, b], (a, b), [a, b) ❛♥❞ (a, b] ❛r❡ t❤❡ ❝❧♦s❡❞✱ ♦♣❡♥ ❛♥❞ ❤❛❧❢✲♦♣❡♥ ✐♥t❡r✈❛❧s ✐♥ R✳ n L ✱ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ Rn ✳ α(n) = Ln {x ∈ Rn : |x| ≤ 1} , t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ✉♥✐t ❜❛❧❧✳ A = ClA✱ t❤❡ ❝❧♦s✉r❡ ♦❢ t❤❡ s❡t A✳ χA , t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ A✳ ✐✐ ♣r♦❞✉❝t x·y ■♥tr♦❞✉❝t✐♦♥ ❈♦✈❡r✐♥❣ t❤❡♦r❡♠ ❤❛s ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ✐♥ t❤❡ st✉❞② ♦❢ ✐♥t❡❣r❛❧s ❛♥❞ ❧✐♠✐ts ♦❢ ✐♥t❡❣r❛❧s✳❚❤❡ t❤❡♦r❡♠ ✇❛s ❢✐rst ❞✐s❝♦✈❡r❡❞ ❛♥❞ ♣r♦✈❡❞ ❜② ●✐✉s❡♣♣❡ ❱✐t❛❧✐ ✐♥ ✶✾✵✽✳ ■t st❛t❡s t❤❛t ✐❢ ❛ s✉❜s❡t ❊ ♦❢ ✐s ❝♦✈❡r❡❞ ❜② ❛ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② ♦❢ s♣❤❡r❡s ✐s ❞r❛✇♥ ❢r♦♠ t❤❛t ❝♦✈❡r✐♥❣ s✉❝❤ t❤❛t t❤❡ ✉♥✐♦♥ ♦❢ t❤♦s❡ ✐s ❝♦✈❡r❡❞ ✇✐t❤ ❊ ◆✱ ✇❤❡r❡ ◆ ✐s ❛ s❡t ♦❢ ▲❡❜❡s❣✉❡ ✵ ♠❡❛s✉r❡✳ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ✐s t❤❡♥ ♣r♦✈❡❞ ❢♦r t❤❡ ❝❛s❡ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡✳ ■t ❛❧s♦ st❛t❡s t❤❛t ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ✐s ❣❡♥❡r❛❧❧② ✐♥❝♦rr❡❝t ❢♦r t❤❡ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s t❤❡s✐s ✐s t♦ ❧❡❛r♥ ❛❜♦✉t ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❛♥❞ s♦♠❡ ♦❢ ✐ts ❛♣♣❧✐❝❛t✐♦♥s✳❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ✐s ❛♥ ✐♥t❡r❡st✐♥❣ r❡s✉❧t ✐♥ t❤❡ ❛♥❛❧②t✐❝❛❧ ❛♥❞ t❤❡♦r❡t✐❝❛❧ t♦♣♦❧♦❣②✳ ❚❤✐s t❤❡s✐s ❝♦♥s✐sts ♦❢ t❤r❡❡ ❝❤❛♣t❡rs✿ ❈❤❛♣t❡r ✶✿ ❲❡ ♣r♦✈❡ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❈❤❛♣t❡r ✷✿ ❲❡ ♣r♦✈❡ ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ♦❢ ❘❛❞♦♥ ♠❡❛s✉r❡s✳ ❈❤❛♣t❡r ✸✿ ❲❡ ❛♣♣❧② t❤❡s❡ ❝♦✈❡r✐♥❣ t❤❡♦r❡♠s t♦ ♣r♦✈❡ s♦♠❡ r❡s✉❧ts ❛❜♦✉t ❞✐❢✲ ❢❡r❡♥t✐❛t✐♦♥ ♦❢ ♠❡❛s✉r❡s✳ ✶ ❈❤❛♣t❡r ✶ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ✺r✲❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❛♥❞ ✉s❡ ✐t t♦ ♣r♦✈❡ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ✶✳✶✳ ❆ ✺r✲❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❚❤❡♦r❡♠ ✶✳✶✳ ❝♦♠♣❛❝t✳ B ▲❡t (X, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t ❛❧❧ ❜♦✉♥❞❡❞ ❝❧♦s❡❞ s✉❜s❡ts ❛r❡ ✐s ❛ ❢❛♠✐❧② ♦❢ ❝❧♦s❡❞ ❜❛❧❧s ✐♥ X s✉❝❤ t❤❛t sup {d(B) : B ∈ B} < ∞ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❢✐♥✐t❡ ♦r ❝♦✉♥t❛❜❧❡ s❡q✉❡♥❝❡ B⊂ ❋✐rst❧②✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t A ♦❢ ❞✐s❥♦✐♥t ❜❛❧❧s s✉❝❤ t❤❛t 5Bi i B∈B Pr♦♦❢✳ Bi ∈ B ✐s ❜♦✉♥❞❡❞ s❡t ❛♥❞ B ❤❛s t❤❡ ❢♦r♠ B = {B(x, r(x)) : x ∈ A} ●✐✈❡♥ M = sup {r(x) : x ∈ A} A1 = ❈❤♦♦s❡ ❛♥ ❛r❜✐tr❛r② x1 ∈ A ❛♥❞ x ∈ A : M < r(x) ≤ M ❛♥❞ t❤❡♥ k xk+1 ∈ A1 \ B(xi , 3r(xi )) i=1 ✷ (1) k ❛s ❧♦♥❣ ❛s A1 \ B(xi , 3r(xi )) = ❈♦♥s✐❞❡r B(xk , r(xk )) ❛♥❞ B(xl , r(xl )) ✇✐t❤ k > l i=1 k−1 xk ∈ A1 \ B(xi , 3r(xi )) ⊂ A \ B(xl , 3r(xl )) i=1 d(xk , xl ) ≥ 3r(xl ) > M = M 4 r(xk ) + r(xl ) ≤ M + M = 2M < ❙♦✱ B(xk , r(xk )) ❛♥❞ B(xl , r(xl )) ❛r❡ ❞✐s❥♦✐♥t✳ ❥♦✐♥t ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❞❡❢✐♥✐t✐♦♥ ♦❢ A1 ✳ ❙♦✱ ❲❡ ❝❤♦♦s❡ t❤❡ ❜❛❧❧s B(xi , r(xi )) ❛r❡ ❞✐s✲ B(xi , r(xi ))✳ ■♥❞❡❡❞✱ ❛ss✉♠❡ t❤❛t A ✐s ❢✐♥✐t❡✳ ❙♦✱ t❤❡r❡ ❡①✐sts A ✐s ❢✐♥✐t❡✿ x1 , , xk+1 , ■t ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ xk ∈ A ✐s ❝♦♥✈❡r❣❡✳ ▼♦r❡♦✈❡r✱ d(xk , xl ) ≤ M ∀k = l ❚❤❡r❡ ❞♦❡s♥✬t ❡①✐sts ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ ❝♦♥tr❛❞✐❝t✐♦♥✮✳ ❙♦✱ ✇❡ ♦♥❧② ❤❛✈❡ ❢✐♥✐t❡❧② ♠❛♥② ♦r t❤❡♠✱ s❛② k1 ✳ ❲❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❡r❡ ✐s ♦♥❧② ❢✐♥✐t❡❧② ♠❛♥② ♦❢ ❚❤✉s✱ ✇❡ ❤❛✈❡ k1 A1 ⊂ B(xi , 3r(xi )) i=1 ❙✐♥❝❡ r(x) ≤ 2r(xi )✱t❤✐s ❣✐✈❡s k1 B(x, r(x)) ⊂ ❢♦r B(xi , 5r(xi )) i=1 x∈A1 x ∈ A1 , i = 1, , ki ●✐✈❡♥ M < r(x) < M A2 = x∈A: A2 = x ∈ A2 : B(x, r(x)) ∩ , k1 B(xi , r(xi )) = ∅ i=1 ■❢ x ∈ A2 \ A2 ✳ ❆s r(x) ≤ 2r(xi ) t❤❡♥ d(x, xi ) ≤ r(x) + r(xi ) ≤ 3r(xi ) k1 ❚❤❡r❡❢♦r❡✱ A2 \ A2 ⊂ B(xi , 3r(xi )) i=1 ❈❤♦♦s❡ ❛♥ ❛r❜✐tr❛r② xki +1 ∈ A2 ❛♥❞ t❤❡♥ k1 xk1 +1 ∈ A2 \ B(xi , 3r(xi )) i=1 ✸ ✭❛ ❙✐♠✐❧❛r②✱ ✇❡ ♦♥❧② ❤❛✈❡ ❢✐♥✐t❡❧② ♠❛♥② ♦❢ B(xi , r(xi )) ❛r❡ ❞✐s❥♦✐♥t✱ s❛② k2 s✉❝❤ t❤❛t A2 ⊂ k2 B(xi , 3r(xi )) i=k1 +1 ❚❤✉s✱ k2 B(x, r(x)) ⊂ i=1 x∈A2 ❢♦r B(xi , 5r(xi )) xi ∈ A2 , i = 1, , k2 Pr♦❝❡❡❞✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r ✇❡ ❢✐♥❞ t❤❡ r❡q✉✐r❡❞ ❜❛❧❧s✳ ■♥ t❤❡ ❛❜♦✈❡ ♣r♦♦❢✱ ✇❡ ❤❛✈❡ t✇♦ r❡str✐❝t✐♦♥s✳ x ∈ A✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❜❛❧❧ B(x, r(x))✳ ❚♦ ❢✐① ✐t✱ ✇❡ 14 sup {r : B(x, r) ∈ B} ❝❛♥ s❡❧❡❝t ❢♦r ❡❛❝❤ ❝❡♥tr❡ x ❛ ❜❛❧❧ B(x, r(x)) ∈ B s✉❝❤ t❤❛t r(x) > 15 ✳ ❛♥❞ ✐♥st❡❛❞ ♦❢ ❝❤♦♦s✐♥❣ ♥✉♠❜❡r ✸ ✐♥ ✭✶✮✱ ✇❡ ✉s❡ ✶✳ ❋✐rst❧②✱ ✇❡ ❛ss✉♠❡❞ t❤❛t ❢♦r ❡❛❝❤ ✷✳ ❙❡❝♦♥❞❧②✱ ✇❡ ❛ss✉♠❡❞ t❤❛t t❤❡ ❝❡♥tr❡s ❧✐❡ ✐♥ ❛ ❜♦✉♥❞❡❞ s❡t✳ ❚♦ ❛✈♦✐❞ t❤✐s t❤❡ ♣r♦♦❢ ❝❛♥ ❜❡ ♠♦❞✐❢✐❡❞ ❜② ❝❤♦♦s✐♥❣ t❤❡ ♥❡✇ ♣♦✐♥ts ❢♦r ❡①❛♠♣❧❡ ✐❢ x ❛♥❞ y xi ♥♦t t♦♦ ❢❛r ❢r♦♠ ❛ ❢✐①❡❞ ♣♦✐♥t ✇❡r❡ ♣♦ss✐❜❧❡ s❡❧❡❝t✐♦♥s ❛♥❞ r✉❧❡ t❤❛t ✇❡ ❝❛♥♥♦t ♣✐❝❦ d(y, a) > 2d(x, a) a ∈ A❀ ✇❡ ✇♦✉❧❞ ♠❛❦❡ ❛ y✳ ❲❡ ❝❛♥ ♥♦✇ ❡❛s✐❧② ❞❡r✐✈❡ ❛ ❱✐t❛❧✐✲t②♣❡ ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛✲ s✉r❡ n L ✶✳✷✳ ✳ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ▲❡❜❡s❣✉❡ ♠❡❛✲ s✉r❡ ❚❤❡♦r❡♠ ✶✳✷✳ ▲❡t A ⊆ Rn ▲❡t B ❜❡ t❤❡ ❢❛♠✐❧② ♦❢ ❝❧♦s❡❞ ❜❛❧❧s ✐♥ ∀x ∈ A : inf {d(B) : x ∈ B ∈ B} = ❚❤❡♥ t❤❡r❡ ❡①✐sts i)Bi1 ∩ Bi2 = ∅ {Bi }∞ i=1 ∈ B ∀i1 = i2 s✉❝❤ t❤❛t ∞ ii)Ln A \ Bi = i=1 ✹ Rn s❛t✐s❢②✐♥❣✿ ❈❤❛♣t❡r ✷ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ❘❛❞♦♥ ♠❡❛s✉r❡s ❚❤❡ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ❘❛❞♦♥ ♠❡❛s✉r❡s ✐s st❛t❡❞ ❛s ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✷✳✶✳ ❆ss✉♠❡ t❤❛t ♦❢ ❝❧♦s❡❞ ❜❛❧❧s ✐♥ R n µ ✐s ❛ ❘❛❞♦♥ ♠❡❛s✉r❡ ♦♥ Rn ✱ A ⊆ Rn ▲❡t B ❜❡ t❤❡ ❢❛♠✐❧② s❛t✐s❢②✐♥❣✿ ∀x ∈ A : inf {r : B(x, r) ∈ B} = {Bi }∞ i=1 s✉❝❤ i)Bi1 ∩ Bi2 = ∅ ∀i1 = i2 ii)µ(A \ Bi ) = ❚❤❡♥ t❤❡r❡ ❡①✐sts t❤❛t✿ i ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ♣r♦✈❡ ❚❤❡♦r❡♠ ✷✳✶ ❜② ✉s✐♥❣ ❇❡s✐❝♦✈✐t❝❤✬s ❝♦✈❡r✐♥❣ t❤❡✲ ♦r❡♠✳ ✷✳✶✳ ❇❡s✐❝♦✈✐t❝❤✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❲❡ s❤❛❧❧ ❜❡❣✐♥ ✇✐t❤ ❛ s✐♠♣❧❡ ❧❡♠♠❛ ❢r♦♠ ♣❧❛♥❡ ❣❡♦♠❡tr②✳ ■♥st❡❛❞ ♦❢ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ❡❧❡♠❡♥t❛r② ❣❡♦♠❡tr✐❝ ❝♦♥s✐❞❡r❛t✐♦♥s ♦♥❡ ❝❛♥ ❛❧s♦ ❡❛s✐❧② ❞❡❞✉❝❡ ✐t ❢r♦♠ t❤❡ ❝♦s✐♥❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛♥❣❧❡ ♦❢ ❛ tr✐❛♥❣❧❡ ✐♥ t❡r♠s ♦❢ t❤❡ s✐❞❡✲❧❡♥❣t❤s✳ ▲❡♠♠❛ ✷✳✶✳ ❆ss✉♠❡ t❤❛t a, b ∈ R2 s❛t✐s❢② t✇♦ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ i, < |a| < |a − b| ii, < |b| < |a − b| ❚❤❡♥ b a − ≥ |a| |b| ✼ Pr♦♦❢✳ ▲❡t a(x1 , y1 ) ❛♥❞ b(x2 , y2 ) ❋r♦♠ t❤❡ ❤②♣♦t❤❡s✐s✱ ✇❡ ❤❛✈❡ (x1 − x2 )2 + (y1 + y2 )2 > x21 + y12 (x1 − x2 )2 + (y1 + y2 )2 > x22 + y22 ■t ❢♦❧❧♦✇s t❤❛t x22 + y22 > 2(x1 x2 + y1 y2 ) x21 + y12 > 2(x1 x2 + y1 y2 ) P✉t (x1 , y1 ) = (r1 cosθ1 , r1 sinθ1 ) (x2 , y2 ) = (r2 cosθ2 , r2 sinθ2 ) ❙♦✱ ✇❡ ❣❡t r2 > 2r1 cos(θ1 − θ2 ) r1 > 2r2 cos(θ1 − θ2 ) ❚❤❡r❡❢♦r❡✱ cos(θ1 − θ2 ) < r2 r1 , 2r1 2r2 ≤ ❲❡ ❤❛✈❡✿ b a − = |a| |b| (cosθ1 − cosθ2 )2 + (sin θ1 − sinθ2 )2 = − 2(cosθ1 cosθ2 + sinθ1 sinθ2 ) = − 2cos(θ1 − θ2 ) ≥ ▲❡♠♠❛ ✷✳✷✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ k ♣♦✐♥ts a1 , , ak ✐♥ Rn ❛♥❞ k ♣♦s✐t✐✈❡ ♥✉♠❜❡rs r1 , , rk s✉❝❤ t❤❛t i)ai ∈ / B(aj , rj ) ❢♦r j = i k ii) B(ai , ri ) = i=1 ❚❤❡♥ k ≤ N (n)✱ ✇❤❡r❡ N (n) ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ ✽ n✳ Pr♦♦❢✳ ❋♦r ❡❛❝❤ i = 1, , k ✱ = ✇❡ s✉♣♣♦s❡ t❤❛t ❛♥❞ k 0∈ B(ai , ri ) i=1 ❲❡ ❤❛✈❡ ∈ B(ai , ri ) ⇒ |ai | < ri ✭✶✮ ∈ / B(aj , rj ) ⇒ |ai − aj | > rj ✭✷✮ ❛♥❞ ❋r♦♠ ✭✶✮ ❛♥❞ ✭✷✮✱ ✇❡ ❤❛✈❡ |ai | < ri < |ai − aj | ❆♣♣❧②✐♥❣ ❧❡♠♠❛ ✷✳✶ ✇✐t❤ ✇❡ ❤❛✈❡ a = ❛♥❞ aj ≥1 − |ai | |aj | S n−1 y1 , , yk ∈ S n−1 b = bj ❢♦r ❢♦r i = j i=j ❢♦r ✐♥ t❤❡ t✇♦✲ ❞✐♠❡♥s✐♦♥❛❧ ♣❧❛♥❡✱ i = j (∗) N (n) ✇✐t❤ i = j ✱ t❤❡♥ k ≤ N (n)✳ ❙✐♥❝❡ t❤❡ ✉♥✐t s♣❤❡r❡ ✐s ❝♦♠♣❛❝t t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r ♣r♦♣❡rt②✿ ✐❢ ✇✐t❤ |yi − yj | ≥ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❇② ✭✯✮✱ N (n) ✐s ✇❤❛t ✇❡ ✇❛♥t✳ ❚❤❡♦r❡♠ ✷✳✷ ✭❇❡s✐❝♦✈✐t❝❤✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠✮✳ Rn ✳ B B✳ ❆ss✉♠❡ t❤❛t ✐s ❛ ❢❛♠✐❧② ♦❢ ❝❧♦s❡❞ ❜❛❧❧s s✉❝❤ t❤❛t ❡❛❝❤ ♣♦✐♥t ♦❢ A A ✐s ❛ ❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✐s t❤❡ ❝❡♥tr❡ ♦❢ s♦♠❡ ❜❛❧❧ ♦❢ ✭✶✮ ❚❤❡r❡ ❡①✐sts ❛ ❢✐♥✐t❡ ♦r ❝♦✉♥t❛❜❧❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❜❛❧❧s Bi ∈ B s✉❝❤ t❤❛t χ(A) ≤ χ(Bi ) ≤ P (n) i ✇❤❡r❡ P (n) ✐s ❛♥ ✐♥t❡❣❡r ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ ♥✳ ✭✷✮ ❚❤❡r❡ ❡①✐sts ❢❛♠✐❧② B1 , , BQ(n) ⊂ B ❝♦✈❡r✐♥❣ A s✉❝❤ t❤❛t Q(n) i, A ⊂ Bi i=1 ii, B ∩ B = ∅ ❢♦r B, B ∈ Bi ✇✐t❤ B = B ✇❤❡r❡ Q(n) ✐s ❛♥ ✐♥t❡❣❡r ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ Pr♦♦❢✳ ✭✶✮ ❋♦r ❡❛❝❤ x ∈ A✱ ♣✐❝❦ ♦♥❡ ❜❛❧❧ ♥✳ B(x, r(x)) ∈ B✳ ❛ss✉♠❡ t❤❛t M1 = sup r(x) < ∞ x∈A ✾ ❆s A ✐s ❜♦✉♥❞❡❞✱ ✇❡ ♠❛② ❈❤♦♦s❡ x1 ∈ A ✇✐t❤ r(x1 ) ≥ M1 ❛♥❞ t❤❡♥ ✐♥❞✉❝t✐✈❡❧② j xj+1 ∈ A \ B(xi , r(xi )) ✇✐t❤ M1 r(xj+1 ) ≥ i=1 ❛s ❧♦♥❣ ❛s ♣♦ss✐❜❧❡✳ ❙✐♥❝❡ s❡q✉❡♥❝❡ A ✐s ❜♦✉♥❞❡❞✱ t❤❡ ♣r♦❝❡ss t❡r♠✐♥❛t❡s✱ ❛♥❞ ✇❡ ❣❡t ❛ ❢✐♥✐t❡ x1 , xk1 ◆❡①t ❧❡t k1 M2 = sup r(x) : x ∈ A \ B(xi , r(xi )) i=1 ❈❤♦♦s❡ k1 xk1 +1 ∈ A \ B(xi , r(xi )) ✇✐t❤ r(xk1 +1 ) ≥ B(xi , r(xi )) ✇✐t❤ r(xj+1 ) ≥ i=1 M2 , ❛♥❞ ❛❣❛✐♥ ✐♥❞✉❝t✐✈❡❧② j xj+1 ∈ A \ i=1 M2 ❈♦♥t✐♥✉✐♥❣ t❤✐s ♣r♦❝❡ss ✇❡ ♦❜t❛✐♥ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs = k0 < k1 < k2 < , ❛ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs Mi ✇✐t❤ 2Mi+1 ≤ Mi ✱ ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ ❜❛❧❧s Bi = B(xi , r(xi ))inB ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✳ ▲❡t Ij = kj−1 + 1, , kj ❢♦r j = 1, 2, Mj ≤ r(xi ) ≤ Mj ❢♦r i ∈ Ij ✭✸✮ Bi ❢♦r j = 1, 2, ✭✹✮ Bj ❢♦r i ∈ Ik ✭✺✮ ❚❤❡♥ j xj+1 ∈ A \ i=1 xi ∈ A \ m=k j∈Im ❚❤❡ ❢✐rst t✇♦ ♣r♦♣❡rt✐❡s ❢♦❧❧♦✇ ✐♠♠❡❞✐❛t❡❧② ❢r♦♠ t❤❡ ❝♦♥str✉❝t✐♦♥✱ ❚♦ ✈❡r✐❢② t❤❡ t❤✐r❞ m = k, j ∈ im ❛♥❞ i ∈ Ik ■❢ m < k, xi ∈ / Bj ❜② ✭✹✮✳ ■❢ k < m, t❤❡♥ r(xj ) < r(xi ), xj ∈ / Bi ❜② ✭✹✮✳ ❙♦✱ xi ∈ / Bj ✳ ❙✐♥❝❡ Mi → 0, ✭✸✮ ✐♠♣❧✐❡s r(xi ) → 0✱ ❛♥❞ ✐t ❢♦❧❧♦✇s ♣r♦♣❡rt②✱ ❧❡t ∞ A⊂ Bi i=1 ✶✵ ❢r♦♠ t❤❡ ❝♦♥str✉❝t✐♦♥ t❤❛t ❚♦ ❡st❛❜❧✐s❤ ❛❧s♦ t❤❡ s❡❝♦♥❞ st❛t❡♠❡♥t ♦❢ ✭✶✮✱ s✉♣♣♦s❡ t❤❛t ❛ ♣♦✐♥t Bi ✱ x ❜❡❧♦♥❣s t♦ p ❜❛❧❧s s❛② p x∈ Bm i=1 ✳ ❲❡ s❤❛❧❧ s❤♦✇ t❤❛t p ≤ P (n) = 16n N (n) ✇✐t❤ N (n) ❛s ✐♥ ▲❡♠♠❛ ✷✳✷✳ ❯s✐♥❣ ✭✺✮ ❛♥❞ ▲❡♠♠❛ ✷✳✷ ✇❡ s❡❡ t❤❛t t❤❡ ✐♥❞✐❝❡s ❞✐❢❢❡r❡♥t ❜❧♦❝❦s Ij ✱ mi ❝❛♥ ❜❡❧♦♥❣ t♦ ❛t ♠♦st N (n) t❤❛t ✐s✱ card {j : Ij ∩ {mi : i = 1, 2, , p} = ∅} ≤ N (n) ❈♦♥s❡q✉❡♥t❧② ✐t s✉❢❢✐❝❡s t♦ s❤♦✇ t❤❛t card (j : Ij ∩ {mi : i = 1, 2, , p}) ≤ 16n ❋✐① j ❢♦r j = 1, 2, ✭✻✮ ❛♥❞ ✇r✐t❡ Ij ∩ {mi : i = 1, 2, , p} = {l1 , , lq } ❇② ✭✸✮ ❛♥❞ ✭✹✮ t❤❡ ❜❛❧❧s B(xli , r(xli )), i = 1, , q, ❛r❡ ❞✐s❥♦✐♥t ❛♥❞ t❤❡② ❛r❡ ❝♦♥t❛✐♥❡❞ B(x, 2Mj )✳ n ❍❡♥❝❡✱ ✇✐t❤ α(n) = L (B(0, 1)), ✐♥ qα(n) Mj q n ≤ i=1 Ln B(xli , r(xli )) ≤ Ln (B(x, 2Mj )) = α(n)(2Mj )n q ≤ 16n ❛s ❞✐s✐r❡❞✳ ❚❤✐s ♣r♦✈❡s ✭✻✮✱ ❛♥❞ t❤✉s ❛❧s♦ ✭✶✮✳ ✭✷✮ ▲❡t B1 , B2 , ❜❡ t❤❡ ❜❛❧❧s ❢♦✉♥❞ ✐♥ ✭✶✮✳ ▲❡tt✐♥❣ Bi = B(xi , ri ), t❤❡r❡ ❛r❡ ❢♦r ❡❛❝❤ > ♦♥❧② ❢✐♥✐t❡❧② ♠❛♥② ❜❛❧❧s Bi ✇✐t❤ ri ≥ ❜❡❝❛✉s❡ ♦❢ ✭✶✮ ❛♥❞ t❤❡ ❜♦✉♥❞❡❞❧❡ss ♦❢ A✳ ❚❤✉s ✇❡ ♠❛② ❛ss✉♠❡ r1 ≥ r2 ≥ ▲❡t B1,1 = B1 ❛♥❞ t❤❡♥ ✐♥❞✉❝t✐✈❡❧② ✐❢ B1,1 , B1,j ❤❛✈❡ ❜❡❡♥ ❝❤♦s❡♥✱ B1,j+1 = Bk ✇❤❡r❡ k ✐s t❤❡ s♠❛❧❧❡st ✐♥t❡❣❡r ✇✐t❤ ❛♥❞ s♦ j Bk ∩ B1,i = ∅ i=1 ❲❡ ❝♦♥t✐♥✉❡ t❤✐s ❛s ❧♦♥❣ ❛s ♣♦ss✐❜❧❡ ❣❡tt✐♥❣ ❛ ❢✐♥✐t❡ ♦r ❝♦✉♥t❛❜❧❡ ❞✐s❥♦✐♥t s✉❜❢❛♠✐❧② B1 = {B1,1 , B1,2 , } ✶✶ B1 , B2 , ■❢ A ✐s ♥♦t ❝♦✈❡r❡❞ ❜② B1 ✱ ✇❡ ❞✐❢✐♥❡❞ ❢✐rst B2,1 = Bk ✇❤❡r❡ k ✐s t❤❡ s♠❛❧❧❡st ✐♥t❡❣❡r ❢♦r ✇❤✐❝❤ Bk ∈ / B1 ✳ ❆❣❛✐♥ ✇❡ ❞❡❢✐♥❡❞ ✐♥❞✉❝t✐✈❡❧② B2,j+1 = Bk ✇✐t❤ s♠❛❧❧❡st k s✉❝❤ t❤❛t ♦❢ j Bk ∩ B2,i = ∅ i=1 B1 , B2 , ✳ ❲✐t❤ t❤✐s ♣r♦❝❡ss ✇❡ ❢✐♥❞ s✉❜❢❛♠✐❧✐❡s ❲❡ ❝❧❛✐♠ t❤❛t ♦❢ B1 , B2 , ✱ ❡❛❝❤ Bi ❜❡✐♥❣ ❞✐s❥♦✐♥t✳ m A⊂ Bk m ≤ 4n P (n) + ❢♦r s♦♠❡ k=1 m ❙✉♣♣♦s❡ m x∈A\ ✐s s✉❝❤ t❤❛t t❤❡r❡ ✐s Bk ❢♦r s♦♠❡ k=1 m ≤ 4n P (n) + m ≤ 4n P (n)✳ ❜❛❧❧s Bi ❝♦✈❡r A✱ ❲❡ t❤❡♥ ❤❛✈❡ t♦ s❤♦✇ t❤❛t x ∈ Bi ✳ ❚❤❡♥ ❢♦r ❡❛❝❤ k = 1, , m, Bi ∈ / Bk , ✇❤✐❝❤ ♠❡❛♥s ❜② t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ Bk t❤❛t Bi ∩ Bk,ik = ∅ ❢♦r s♦♠❡ ik ❢♦r ✇❤✐❝❤ ri ≤ rk,ik , ri ❛♥❞ rk,ik ❜❡✐♥❣ t❤❡ r❛❞✐✐ ♦❢ Bi ❛♥❞ Bk,ik , r❡s♣❡❝t✐✈❡❧②✳ ri ❝♦♥t❛✐♥❡❞ ✐♥ (2Bi ) ∩ Bk,ik ❢♦r ❛❧❧ k = 1, , m ❍❡♥❝❡✱ t❤❡r❡ ❛r❡ ❜❛❧❧s Bk ♦❢ r❛❞✐✉s n ❙✐♥❝❡ ❡❛❝❤ ♣♦✐♥t ♦❢ R ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛t ♠♦st P (n) ❜❛❧❧s Bk,ik , k = 1, , m, t❤✐s ✐s ❛❧s♦ tr✉❡ ❢♦r t❤❡ s♠❛❧❧❡r ❜❛❧❧s Bk ✱ t❤❛t ✐s ❉✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ✇❡ ❝❛♥ ❢✐♥❞ i ✇✐t❤ m χBk ≤ P (n)χ∪m k=1 Bk k=1 ❯s✐♥❣ t❤❡ ❢❛❝t Bk ⊂ 2Bi ✱ t❤❡♥ ✇❡ ❤❛✈❡ 2n α(n)rin = Ln (2Bi ) m n ≥L Bk k=1 = χ∪m dLn k=1 Bk m −1 χBk dLn ≥ P (n) k=1 m = P (n)−1 Ln (Bk ) k=1 −1 −n = mP (n) ❍❡♥❝❡✱ m ≤ 4n P (n) ❛s r❡q✉✐r❡❞✳ ✶✷ α(n)rin ✷✳✷✳ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✶ µ(A) > 0✳ ❋✐rst❧②✱ n ♠❡❛s✉r❡ ♦♥ R , t❤❡♥ ❙✉♣♣♦s❡ t❤❛t µ ✐s ❛ ❘❛❞♦♥ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ❆ ✐s ❜♦✉♥❞❡❞✳ µ(A) = inf {µ(U ) : A ⊂ U, U ■t ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ❛♥ ♦♣❡♥ s❡t U s✉❝❤ t❤❛t µ(U ) ≤ ✇❤❡r❡ Q(n) ✐s ♦♣❡♥ A⊂U 4Q(n) 1+ } ❢♦r A ⊂ Rn ❛♥❞ µ(A) ✐s ❛s ✐♥ ❇❡s✐❝♦✈✐t❝❤✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠✳ B1 , , BQ (n) ∈ B ❆s ❛ r❡s✉❧t ♦❢ t❤❛t t❤❡♦r❡♠✱ ✇❡ ❝❛♥ ❢✐♥❞ B; Bi1 ∩ Bi2 = ∅ ∀i1 = i2 s✉❝❤ t❤❛t ❛♥❞ Q(n) A⊂ Bi ⊂ U i=1 ❙♦✱ µ(A) ≤ µ Bi Q(n) i=1 Q(n) = Bi µ i=1 ❙♦✱ t❤❡r❡ ❡①✐sts ❛♥ i s✉❝❤ t❤❛t µ(A) ≤ Q(n).µ ❋✉rt❤❡r✱ ❢♦r s♦♠❡ ❢✐♥✐t❡ s✉❜❢❛♠✐❧② µ(A) ≤ 2Q(n).µ P✉t A1 = A \ Bi ♦❢ Bi ✱ Bi ✇❡ ❤❛✈❡ Bi ⇒ µ Bi ≥ µ(A) 2Q(n) Bi ❲❡ ♦❜t❛✐♥ µ(A1 ) = µ A \ Bi ≤µ U\ Bi = µ(U ) − µ ≤ = Bi 1 − 4Q(n) 2Q(n) 1− µ(A) 4Q(n) 1+ = u.µ(A) ✶✸ µ(A) {Bi }Q(n) i=1 ⊂ ✇❤❡r❡ ◆♦✇ u= A1 1− 4Q(n) < 1, Q(n) > U1 ✐s ❜♦✉♥❞❡❞ s❡t ❛♥❞ ❝❤♦♦s❡ s✉❝❤ t❤❛t Q(n) A1 ⊂ U1 ⊆ U \ Bi ❛♥❞ i=1 µ(U1 ) ≤ 1+ 4Qnr(n) µ(A1 ) ❆s ❛❜♦✈❡✱ ✇❡ ❤❛✈❡ µ(A2 ) ≤ 1− 4Q(n) ≤ 1− 4Q(n) µ(A1 ) µ(A) = u2 µ(A) ✇❤❡r❡ A2 = A1 \ Bi , Bi ✐s ❢✐♥✐t❡ s✉❜❢❛♠✐❧② ♦❢ Bi ✳ Pr♦❝❡❡❞✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ✇❡ ❣❡t µ A\ Bi ≤ um µ(A) i ✇❤❡r❡ ❚❤✉s✱ µ A\ Bi =0 ❛s m → ∞ i ✶✹ u=1− < 4Q(n) ❈❤❛♣t❡r ✸ ❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✉s❡ ❝♦✈❡r✐♥❣ t❤❡♦r❡♠s t♦ st✉❞② t❤❡ ❞✐❢❢❡r❡♥t✐❛❧ t❤❡♦r② ♦❢ ♠❡❛s✉r❡s✳ ❉❡❢✐♥✐t✐♦♥ ✸✳✶✳ ❆ss✉♠❡ t❤❛t µ ❛♥❞ λ ❛r❡ ❧♦❝❛❧❧② ❢✐♥✐t❡ ❇♦r❡❧ ♠❡❛s✉r❡s ♦♥ ❚❤❡♥ D(µ, λ, x) = lim sup r↓0 D(µ, λ, x) = lim inf r↓0 Rn ✳ µ(B(x, r)) λ(B(x, r)) µ(B(x, r)) λ(B(x, r)) ❛r❡ ❝❛❧❧❡❞ r❡s♣❡❝t✐✈❡❧② t❤❡ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❞❡r✐✈❛t✐✈❡s ♦❢ µ ✇✐t❤ r❡s♣❡❝t t♦ λ ❛t ❛ ♣♦✐♥t n x∈R ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ µ ❡①✐sts ❛t t❤❡ ♣♦✐♥t x ✐❢ ❛♥❞ ♦♥❧② ✐❢ D(µ, λ, x) = D(µ, λ, x) = D(µ, λ, x) ❉❡❢✐♥✐t✐♦♥ ✸✳✷✳ ■❢ µ ❛♥❞ t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ λ λ ❛r❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Rn ✳ ❚❤❡♥ µ ✐s ❝❛❧❧❡❞ ❛❜s♦❧✉t❡❧② ❝♦♥✲ ✐❢ µ(A) = ❚❤❡ ❛❜s♦❧✉t❡ ❝♦♥t✐♥✉✐t② ♦❢ µ ❢♦r ❛♥② A ⊂ Rn ✇✐t❤ r❡s♣❡❝t t♦ ■♥ t❤❡ ♦t❤❡r✇♦r❞✱ t❤❡ ♣r♦♣❡rt② µ λ λ s✉❝❤ t❤❛t λ(A) = ✐s ❞❡♥♦t❡❞ ❜② µ λ✳ ✐s ❡q✉❛✈❛❧❡♥t t♦ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿ ❢♦r ε > 0, t❤❡r❡ ✐s ❛ δ > s✉❝❤ t❤❛t µ(A) < ε ❢♦r ❡✈❡r② A ⊂ Rn ✇✐t❤ λ(A) < δ ❛♥② ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ♣❧❛②s t❤❡ ❦❡② r♦❧❡ ❢♦r t❤❡ ♣r♦♦❢ ♦❢ ♦✉r ♠❛✐♥ r❡s✉❧t ✭❚❤❡♦r❡♠ ✸✳✶✮✳ ✶✺ ▲❡t ▲❡♠♠❛ ✸✳✶✳ A ❜❡ ❛ s✉❜s❡t ♦❢ Rn ❛♥❞ µ, λ ❜❡ ❘❛❞♦♥ ♠❡❛s✉r❡s ♦♥ ❡❛❝❤ 00 {x : ∃D(µ, λ, x) < ∞} ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts As,t,r ❛♥❞ ✇❤❡r❡ s ❛♥❞ t r✉♥ t❤r♦✉❣❤ t❤❡ ♣♦s✐t✐✈❡ r❛t✐♦♥❛❧s ✇✐t❤ s < t ❛♥❞ r r✉♥s t❤r♦✉❣❤ ❇✉t t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ t❤❡ s❡t Au,r u>0 t❤❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳ ❍❡♥❝❡ ✐t ✐s ♦❢ ❚♦ ♣r♦✈❡ (ii) ❝❤♦♦s❡ 1