❍❆◆❖■ 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 1

❛♥❞

3
x ∈ A : M < r(x) ≤ M
4

.

❛♥❞ t❤❡♥

k

xk+1 ∈ A1 \

B(xi , 3r(xi ))
i=1

✷

(1)



k
❛s ❧♦♥❣ ❛s

A1 \

B(xi , 3r(xi )) = 0. ❈♦♥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

3
9
d(xk , xl ) ≥ 3r(xl ) > 3. M = M
4
4
9
r(xk ) + r(xl ) ≤ M + M = 2M < .
4
❙♦✱ 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❣❡✳
9
▼♦r❡♦✈❡r✱ d(xk , xl ) ≤
M ∀k = l. ❚❤❡r❡ ❞♦❡s♥✬t ❡①✐sts ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡
4
❝♦♥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 .

●✐✈❡♥

3
4

2

3
M < r(x) < M
4

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
8
✳
❛♥❞ ✐♥st❡❛❞ ♦❢ ❝❤♦♦s✐♥❣ ♥✉♠❜❡r ✸ ✐♥ ✭✶✮✱ ✇❡ ✉s❡

3

✶✳ ❋✐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} = 0.
❚❤❡♥ t❤❡r❡ ❡①✐sts

i)Bi1 ∩ Bi2 = ∅


{Bi }∞
i=1 ∈ B
∀i1 = i2

s✉❝❤ t❤❛t

∞

ii)Ln A \

Bi

= 0.

i=1

✹

Rn

s❛t✐s❢②✐♥❣✿


▼♦r❡♦✈❡r✱ ❢♦r ❡✈❡r②

ε > 0✱

✇❡ ❝❛♥ ❝❤♦♦s❡


Bi

s✉❝❤ t❤❛t

∞

Ln (Bi ) ≤ Ln (A) + ε.
i=1

Pr♦♦❢✳

❆ss✉♠❡ t❤❛t

A

✐s ❜♦✉♥❞❡❞✳ ▲❡t

V

❜❡ ❛♥ ♦♣❡♥ s❡t s✉❝❤ t❤❛t

A⊂V

Ln (V ) ≤ (1 + 7−n ).Ln A.
∃Bi ∈ B, i = 1, .., k,
∀i1 = i2 ; Bi ⊂ V

❆♣♣❧②✐♥❣ ❛ ✺r✲❝♦✈❡r✐♥❣ t❤❡♦r❡♠✱

Bi1 ∩ Bi2 = ∅

A⊂
B(xi , 5ri ). ❚❤❡♥

s✉❝❤ t❤❛t
❛♥❞

i

5−n Ln (A) ≤ 5−n

Ln (B(xi , 5ri )) =5−n .Cn .(5ri )n
i

=Cn .rin
Ln (Bi )

=
i
k

Ln (Bi ) ≥ 5−n Ln (A)

⇔ lim

k→+∞

i=1

n


L (A) = a > 0

■❢

k

Ln (Bi ) ≥ 5−n a > 0

lim

k→∞

i=1
k1

❚❤❡♥

Ln (Bi ) ≥ 5−n a − ε.

∀ε > 0, ∃k1 :
i=1

❈❤♦♦s❡

ε = (5−n − 6−n )a > 0

❙♦✱

k1


Ln (Bi ) ≥ 6−n a = 6−n Ln (A).
i=1
k1
▲❡tt✐♥❣

A1 = A \

k1

Bi ⊆ V \
i=1

Bi
i=1
k1

n

n

k1
n

i=1

i=1
−n

✇❤❡r❡


Ln (Bi )

Bi ) =L (V ) −

L (A1 ) ≤ L (V \

=(1 + 7 )Ln (A) − 6−n Ln (A)
1
1
=(1 + n − n )Ln (A)
7
6
n
=u.L (A)
1
1
u = 1 + n − n.
7
6

✺

❛♥❞


◆♦✇

A1

✐s ❜♦✉♥❞❡❞ s❡t ❛♥❞ ❝❤♦♦s❡


V1

s✉❝❤ t❤❛t

k1

A1 ⊆ V1 ⊆ V \

Bi
i=1

❛♥❞

❆♥❞ s✐♠✐❧❛r②✱

L (V1 ) ≤ (1 + 7−n )Ln (A1 ).
n

∃Bi ∈ B, i = k1 + 1, .., k2

s✉❝❤ t❤❛t

Bi1 ∩ Bi2 = ∅

∀i1 = i2

❛♥❞

Ln (A2 ) ≤ u.Ln (A1 ) ≤ u2 .Ln (A)

k2
✇❤❡r❡

k2

A2 = A1 \

Bi = A \

❛❧❧ t❤❡ ❜❛❧❧s

Bi ,

i=1

i=k1 +1

i = 1, .., k2

Bi ,

❛r❡ ❞✐s❥♦✐♥t✳

Pr♦❝❡❡❞✐♥❣ ✐♥ t❤✐s ♠❛♥♥❡r✱ ✇❡ ❣❡t t❤❛t

km

Bi ) ≤ um .Ln (A)

n


L (A \
i=1

✇❤❡r❡

1
1
− n < 1.
n
7
6

u=1+

km

❚❤❡r❡❢♦r❡✱

n

L (A \

Bi ) = 0

❛s

m→∞

i=1

♦r

Ln (A \

Bi ) = 0.
i=1

❘❡♠❛r❦ ✶✳✶✳

■♥ t❤❡ t❤❡♦r❡♠ ✶✳✷✱ ✐❢ ✇❡ r❡♣❧❛❝❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡

Ln

❜② t❤❡ ❣❡♥❡r❛❧

❘❛❞♦♥ ♠❡❛s✉r❡ t❤❡♥ t❤❡ r❡s✉❧t ❞♦❡s ♥♦t ❤♦❧❞✳
❊①❛♠♣❧❡ ✶✳✶✳

❆ss✉♠❡ t❤❛t

µ

✐s ❛ ❘❛❞♦♥ ♠❡❛s✉r❡ ♦♥

R2

❞❡❢✐♥❡❞ ❜②

µ(A) = L1 ({x ∈ R : (x, 0) ∈ A}) .
t❤❛t ✐s✱


µ

✐s t❤❡ ❧❡♥❣t❤ ♠❡❛s✉r❡ ♦♥ t❤❡

x✲❛①✐s✳

❚❤❡ ❢❛♠✐❧②

B = {B((x, y), y) : x ∈ R, 0 < y < ∞} .
❝♦✈❡rs

A = {(x, 0) : x ∈ R}
❧❡❝t✐♦♥ B1 , B2 , ... ✇❡ ❤❛✈❡

✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡♦r❡♠ ✶✳✷ ❜✉t ❢♦r ❛♥② ❝♦✉♥t❛❜❧❡ s✉❜❝♦❧✲
∞

µ A∩

Bi

= 0.

i=1
❚❤❡r❡ ✐s ❛ ✈❡rs✐♦♥ ♦❢ ❱✐t❛❧✐✬s ❝♦✈❡r✐♥❣ t❤❡♦r❡♠ ❢♦r ❘❛❞♦♥ ♠❡❛s✉r❡ ❜✉t ✐t r❡q✉✐r❡s
♠♦r❡ ❝♦♥❞✐t✐♦♥s✳ ❚❤❛t t❤❡♦r❡♠ ✇✐❧❧ ❜❡ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳

✻



❈❤❛♣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} = 0.
{Bi }∞
i=1 s✉❝❤
i)Bi1 ∩ Bi2 = ∅
∀i1 = i2
ii)µ(A \

Bi ) = 0.
❚❤❡♥ 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, 0 < |a| < |a − b|
ii, 0 < |b| < |a − b|
❚❤❡♥

b

a
−
≥ 1.
|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 ) < min

r2 r1
,
2r1 2r2

1
≤ .
2

❲❡ ❤❛✈❡✿

b
a
−
=
|a| |b|

(cosθ1 − cosθ2 )2 + (sin θ1 − sinθ2 )2

=

2 − 2(cosθ1 cosθ2 + sinθ1 sinθ2 )

=


2 − 2cos(θ1 − θ2 )

≥ 1.

▲❡♠♠❛ ✷✳✷✳

❆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 ) = 0.
i=1

❚❤❡♥

k ≤ N (n)✱

✇❤❡r❡

N (n)

✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥

✽

n✳


Pr♦♦❢✳

❋♦r ❡❛❝❤

i = 1, ..., k ✱

ai = 0

✇❡ s✉♣♣♦s❡ t❤❛t


❛♥❞

k

0∈

B(ai , ri ).
i=1

❲❡ ❤❛✈❡

0 ∈ B(ai , ri ) ⇒ |ai | < ri

✭✶✮

ai ∈
/ B(aj , rj ) ⇒ |ai − aj | > rj

✭✷✮

❛♥❞

❋r♦♠ ✭✶✮ ❛♥❞ ✭✷✮✱ ✇❡ ❤❛✈❡

|ai | < ri < |ai − aj |
❆♣♣❧②✐♥❣ ❧❡♠♠❛ ✷✳✶ ✇✐t❤
✇❡ ❤❛✈❡

a = ai


❛♥❞

aj
ai
≥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 | ≥ 1

❢♦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
2

❛♥❞ t❤❡♥ ✐♥❞✉❝t✐✈❡❧②

j

xj+1 ∈ A \

B(xi , r(xi ))

✇✐t❤

M1

2

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
,
2

❛♥❞ ❛❣❛✐♥ ✐♥❞✉❝t✐✈❡❧②

j

xj+1 ∈ A \
i=1

M2
.
2

❈♦♥t✐♥✉✐♥❣ t❤✐s ♣r♦❝❡ss ✇❡ ♦❜t❛✐♥ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs


0 = 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
2

❢♦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

1
B(xli , r(xli )), i = 1, ..., q,
4

❛r❡ ❞✐s❥♦✐♥t ❛♥❞ t❤❡② ❛r❡ ❝♦♥t❛✐♥❡❞

B(x, 2Mj )✳
n
❍❡♥❝❡✱ ✇✐t❤ α(n) = L (B(0, 1)),
✐♥

qα(n)

Mj
8

q

n

≤
i=1

1

Ln B(xli , r(xli ))
4

≤ 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 ❡❛❝❤
> 0 ♦♥❧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) + 1.

❢♦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) + 1.

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
2
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) 2
❍❡♥❝❡✱

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

1
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 ≥

1
.µ(A)
2Q(n)

Bi

❲❡ ♦❜t❛✐♥

µ(A1 ) = µ A \

Bi

≤µ U\

Bi

= µ(U ) − µ
≤
=


Bi

1
1
−
4Q(n) 2Q(n)
1
1−
µ(A)
4Q(n)
1+

= u.µ(A)

✶✸

µ(A)

{Bi }Q(n)
i=1 ⊂


✇❤❡r❡
◆♦✇

u=

A1


1−

1
4Q(n)

< 1, Q(n) > 0.
U1

✐s ❜♦✉♥❞❡❞ s❡t ❛♥❞ ❝❤♦♦s❡

s✉❝❤ t❤❛t

Q(n)

A1 ⊂ U1 ⊆ U \

Bi

❛♥❞

i=1

µ(U1 ) ≤

1+

1
4Qnr(n)

µ(A1 )


❆s ❛❜♦✈❡✱ ✇❡ ❤❛✈❡

µ(A2 ) ≤

1−

1
4Q(n)

≤

1−

1
4Q(n)

µ(A1 )
2

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−

1
< 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) = 0
❚❤❡ ❛❜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) = 0.

✐s ❞❡♥♦t❡❞ ❜②

µ

λ✳


✐s ❡q✉❛✈❛❧❡♥t t♦ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿ ❢♦r

ε > 0, t❤❡r❡ ✐s ❛ δ > 0 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 ♦♥

❡❛❝❤

0

i)■❢ D(µ, λ, x) ≤ t ❢♦r ❛❧❧ x ∈ A, t❤❡♥ µ(A) ≤ tλ(A),
ii) ■❢ D(µ, λ, x) ≥ t ❢♦r ❛❧❧ x ∈ A, t❤❡♥ µ(A) ≥ tλ(A).
Pr♦♦❢✳

(i)λ

✐s ❘❛❞♦♥ ♠❡❛s✉r❡✳ ❙♦✱ ✇❡ ❝❛♥ ❢✐♥❞ ❛♥ ♦♣❡♥ s❡t ❯ s✉❝❤ t❤❛t

λ(A) = inf {λ(U ) : A ⊂ U }
∃δ > 0 : λ(U ) ≤ λ(A) + δ ✳
❚❤❡♦r❡♠ ✷✳✶✱ ∃Bi ⊂ U, Bi1 ∩ Bi2 = ∅

■t ✐♠♣❧✐❡s t❤❛t
❆♣♣❧✐♥❣

µ(A \

❢♦r

i1 = i2

❛♥❞ s❛t✐s❢②

Bi ) = 0
i

❋r♦♠ t❤❡ ❤②♣♦t❤❡s✐s✱

D(µ, λ, x) ≤ t✱

t❤❡♥

D(µ, λ, x) = lim sup
r↓0

❙❡t

µ(B(x, r))
≤t
λ(B(x, r))

µ(Bi ) ≤ (t + δ)λ(Bi ).

❲❡ ❤❛✈❡

µ(A) ≤

µ(Bi )
i

≤

(t + δ)λ(Bi )
i

≤ (t + δ)λ(V )
≤ (t + δ)(λ(A) + δ)
▲❡tt✐♥❣

(ii)

δ ↓ 0✱

✇❡ ✇✐❧❧ ♦❜t❛✐♥

µ(A) ≤ tλ(A).

❲❡ ❤❛✈❡

µ(B(x, r))
≥t
r↓0
λB(x, r)
λ(B(x, r))
1
⇔ lim sup
≤
µB(x, r)
t
r↓0
1
⇔ D(λ, µ, x) ≤ .
t

D(µ, λ, x) ≥ t ⇔ lim inf

❚❤❡♥✱ ❜② t❤❡ ♣❛rt

(i)✱

✇❡ ❤❛✈❡

1
λ(A) ≤ µ(A).
t
❍❡♥❝❡

µ(A) ≥ tλ(A).

✶✻

Rn ✳

❚❤❡♥✱ ❢♦r


❚❤❡♦r❡♠ ✸✳✶✳

❆ss✉♠❡ t❤❛t

µ

❛♥❞

λ

❛r❡ ❘❛❞♦♥ ♠❡❛s✉r❡s ♦♥

Rn .

❚❤❡♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣

st❛t❡♠❡♥ts ❛r❡ ❢✉❧❢✐❧❧❡❞✿

i)❚❤❡ ❞❡r✐✈❛t✐✈❡ D(µ, λ, x) ❛❧✇❛②s ❡①✐sts ❢♦r λ
x ∈ Rn . ▼♦r❡♦✈❡r✱ D(µ, λ, x) < +∞✳
ii) ❋♦r ❡❛❝❤ ❇♦r❡❧ s❡t B ⊂ Rn ✱ t❤❡ ✐♥❡q✉❛❧✐t②

❛❧♠♦st ❛❧❧

D(µ, λ, x)dλx ≤ µ(B)
B

❤♦❧❞s ♣r♦✈✐❞❡❞ t❤❛t

iii)µ
Pr♦♦❢✳

λ

µ

✐❢ ❛♥❞ ♦♥❧② ✐❢

❋♦r ❡❛❝❤

λ.
D(µ, λ, x) < ∞

❢♦r

µ

0 < r < ∞, 0 < s < t < ∞✱

❛❧♠♦st ❛❧❧

x ∈ Rn .

❧❡t

As,t,r = x ∈ B(r) : D(µ, λ, x) ≤ s ≤ t ≤ D(µ, λ, x)
❛♥❞

At,r = x ∈ B(r) : D(µ, λ, x) ≥ t
❆s ❛ r❡s✉❧t ♦❢ ❧❡♠♠❛ ✸✳✶✱ ✇❡ ❤❛✈❡

tλ(As,r,t ) ≤ µ(As,t,r ) ≤ sλ(As,t,r ) < ∞
❛♥❞

uλ(Au,r ) ≤ µ(Au,r ) ≤ µ(B(r)) < ∞
❚❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s ②❡✐❧❞

λ(As,t,r ) = 0
λ(

s✐♥❝❡

s < t✱

❛♥❞

Au,r ) = lim λ(Au,r ) = 0.
u→∞

u>0

{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
λ

♠❡❛s✉r❡s ③❡r♦✱ ✇❤✐❝❤ s❡tt❧❡s

(i)✳

❛♥❞ ❧❡t

Bp = x ∈ B : tp ≤ D(µ, λ, x) < tp+1 , p = 0, ±1, ±2, ...
❚❤❡♥ ❜② ♣❛rt

(i) ♦❢ t❤✐s t❤❡♦r❡♠ ❛❧r❡❛❞② ♣r♦✈❡❞ ❛♥❞ ❜② ♣❛rt (ii) ♦❢ ❧❡♠♠❛ ✸✳✶✱✇❡ ❤❛✈❡
∞

D(µ, λ, x)dλx =

D(µ, λ, x)dλx
p=−∞B
p

B

∞

tp+1 λ(Bp )

≤
p=−∞
∞

≤t

µ(Bp )
p=−∞

≤ tµ(B).

✶✼


λ ✐s ❛ ❘❛❞♦♥ ♠❡❛s✉r❡ ♦♥ Rn .
i, ■❢ A ✐s ❛ λ✲♠❡❛s✉r❡ s❡t ♦♥Rn t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❤♦❧❞s
λ(A ∩ B(x, r)) 1 ❢♦r λ ❛❧♠♦st ❛❧❧ x ∈ A
lim
=
r↓0
0 ❢♦r λ ❛❧♠♦st ❛❧❧ x ∈ Rn \ A.
λ(B(x, r))
ii, ■❢ f : Rn → R ✐s ❧♦❝❛❧❧② λ ✐♥t❡❣r❛❜❧❡✱ t❤❡♥
❆ss✉♠❡ t❤❛t

❈♦r♦❧❧❛r② ✸✳✶✳

1
λ(B(x, r))

lim
r↓0

f dλ = f (x)
B(x,r)

❢♦r

λ

❛❧♠♦st ❛❧❧②

Pr♦♦❢✳

x ∈ Rn .

❲❡ ✇✐❧❧ ♣r♦✈❡

❆ss✉♠❡ t❤❛t

f ≥ 0✳

(ii)

❢✐rst✳

❉❡❢✐♥❡ t❤❡ ❘❛❞♦♥ ♠❡❛s✉r❡

µ

❜②

µ(A) =

f dλ.

❚❤❡♥

µ

λ

❛♥❞

A
❜② t❤❡♦r❡♠ ✸✳✶✱ ✇❡ ❤❛✈❡

D(µ, λ, x)dλx = µ(B) =
B

B

❢♦r ❛❧❧ ❇♦r❡❧ s❡ts ❇✳ ■t ✐♠♣❧✐❡s t❤❛t

(ii)✳
♣r♦✈❡ (i)✱

f dλ

f (X) = D(µ, λ, x)

❢♦r

λ

❛❧♠♦st ❛❧❧

x ∈ Rn ,

✇❤✐❝❤

♣r♦✈❡s
❚♦

✇❡ s✉❜st✐t✉❡

❉❡❢✐♥✐t✐♦♥ ✸✳✸✳

A⊂R

n

f = χ(A).

❆ss✉♠❡ t❤❛t

µ

❛♥❞

λ

❛r❡ ❘❛❞♦♥ ♠❡❛s✉r❡s ♦♥

Rn ✳

■❢ t❤❡r❡ ❡①✐st ❛ s❡t

s✉❝❤ t❤❛t

λ(A) = 0 = µ(Rn \ A)
t❤❡♥

µ

❛♥❞

λ

❛r❡ ❝❛❧❧❡❞ ♠✉t✉❛❧❧② s✐♥❣✉❧❛r✳

❚❤❡♦r❡♠ ✸✳✷✳

❢✉♥❝t✐♦♥

▲❡t

µ

❛♥❞

λ

❜❡ ❢✐♥✐t❡ ❘❛❞♦♥ ♠❡❛s✉r❡s ♦♥

❛♥❞ ❛ ❘❛❞♦♥ ♠❡❛s✉r❡

f

µ(B) =

ν

s✉❝❤ t❤❛t

f dλ + ν(B)

λ

❛♥❞

ν

Rn ✳

❛r❡ ♠✉t✉❛❧❧② s✐♥❣✉❧❛r ❛♥❞

❢♦r ❇♦r❡❧ s❡ts

B ⊂ Rn .

B

▼♦r❡♦✈❡r✱
Pr♦♦❢✳

µ

λ

✐❢ ❛♥❞ ♦♥❧② ✐❢

ν = 0.

P✉t

A = {x ∈ Rn : D(µ, λ, x) < ∞} ,
❛♥❞

µ1 A

❛♥❞

ν = µ (Rn \ A).

❚❤❡♥ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t

µ = µ1 + ν,

✶✽

❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❇♦r❡❧


❛♥❞

λ

ν

❛♥❞

❛r❡ ♠✉t✉❛❧❧② s✐♥❣✉❧❛r ❜② t❤❡♦r❡♠ ✸✳✶✳

▼♦r❡♦✈❡r✱ ▲❡♠♠❛ ✸✳✶ ❣✐✈❡s

µ1

λ

❤❛s ❞❡s✐r❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ❜② t❤❡♦r❡♠ ✸✳✶ ✇✐t❤

f = D(µ1 , λ, ).
❙♦

µ(B) =

f dλ + ν(B)

❢♦r ❇♦r❡❧ s❡ts

B ⊂ Rn .

B
❚❤❡ ❧❛st st❛t❡♠❡♥t ✐s ♦❜✈✐♦✉s✳

❈♦r♦❧❧❛r② ✸✳✷✳

✭ ❚❤❡ ❘❛❞♦♥✲ ◆✐❦♦❞②♠ t❤❡♦r❡♠ ❢♦r ❢✐♥✐t❡ ❘❛❞♦♥ ♠❡❛s✉r❡s✮ ❆ss✉♠❡

µ ❛♥❞ λ ❛r❡ ❢✐♥✐t❡ ❘❛❞♦♥ ♠❡❛s✉r❡s ♦♥ Rn ✳ ■❢ µ
λ ✭✐✳❡✱ µ ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s
✇✐t❤ r❡s♣❡❝t t♦ λ✮✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f : Rn → [0, ∞) s✉❝❤ t❤❛t ❢♦r ❛♥②
❇♦r❡❧ s❡t A ⊆ Rn ✱ ✇❡ ❤❛✈❡

t❤❛t

µ(A) =

f dλ.
A

❋✉t❤❡r ❢ ✐s ✉♥✐q✉❡ ❛❧♠♦st ❡✈❡r②✇❤❡r❡ r❡❧❛t❡❞ t♦

µ(A) =

hdλ,

t❤❡♥

λ✱

✐✳❡✱ ✐❢

λ−

f =h

❛❧♠♦st ❡✈❡r②✇❤❡r❡✳

A

❚❤❡ ❢✉♥❝t✐♦♥

f

❈♦r♦❧❧❛r② ✸✳✸✳

❆ss✉♠❡ t❤❛t

µ

✐s ❝❛❧❧❡❞ t❤❡ ❘❛❞♦♥✲◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ ❛♥❞ ✐s ❞❡♥♦t❡❞ ❜②

✭▲❡❜❡❣✉❡ ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡♦r❡♠ ❢♦r ❢✐♥✐t❡ ❘❛❞♦♥ ♠❡❛s✉r❡s✮
❛♥❞

σ ✲❢✐♥✐t❡ ♠❡❛s✉r❡s µs
i, µa
λ, µs λ
ii, µ = µa + µs .
❊①❛♠♣❧❡ ✸✳✶✳

▲❡t

λ = L
. ❚❤❡♥
0
❢ ❂

e−a

λ

❛r❡ t✇♦ ❢✐♥✐t❡ ❘❛❞♦♥ ♠❡❛s✉r❡s ♦♥

❛♥❞

µa

0

1 − e−a

✐❢

✐❢

R

❞❡❢✐♥❡❞ ❜②✿

a<0

a≥0

t❤❡ ❘❛❞♦♥✲ ◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ ✐s✿
✐❢

a<0

✐❢

a ≥ 0.

Rn ✳

s✉❝❤ t❤❛t

▲❡tµ ❜❡ ❛ ❘❛❞♦♥ ♠❡❛s✉r❡ ♦♥

µ[−∞, a] =
1

dµ
✳
dλ

✶✾

❚❤❡♥ t❤❡r❡ ❡①✐sts t✇♦