❍❆◆❖■ P❊❉❆●❖●■❈❆▲ ❯◆■❱❊❘❙■❚❨ ✷
❋❆❈❯▲❚❨ ❖❋ ▼❆❚❍❊▼❆❚■❈❙
✖✖✖✖♦✵♦✖✖✖✖

❉❖❆◆ ❚❍■ P❍❯❖◆●

❍❆❯❙❉❖❘❋❋ ▼❊❆❙❯❘❊❙ ❆◆❉ ❉■▼❊◆❙■❖◆

❇❆❈❍❊▲❖❘ ❚❍❊❙■❙

▼❛❥♦r✿ ❆♥❛❧②s✐s

❍❛♥♦✐ ✕ ✷✵✶✾


❍❆◆❖■ P❊❉❆●❖●■❈❆▲ ❯◆■❱❊❘❙■❚❨ ✷
❋❆❈❯▲❚❨ ❖❋ ▼❆❚❍❊▼❆❚■❈❙
✖✖✖✖♦✵♦✖✖✖✖

❉❖❆◆ ❚❍■ P❍❯❖◆●

❍❆❯❙❉❖❘❋❋ ▼❊❆❙❯❘❊❙ ❆◆❉ ❉■▼❊◆❙■❖◆

❇❆❈❍❊▲❖❘ ❚❍❊❙■❙

▼❛❥♦r✿ ❆♥❛❧②s✐s

❙✉♣❡r✈✐s♦r✿

❉r✳ ❉♦ ❍♦❛♥❣ ❙♦♥

❍❛♥♦✐ ✕ ✷✵✶✾


❆❝❦♥♦✇❧❡❞❣♠❡♥t
■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② ❣r❛t✐t✉❞❡ t♦ t❤❡ t❡❛❝❤❡r ♦❢ t❤❡ ❋❛❝✉❧t② ♦❢
▼❛t❤❡♠❛t✐❝s✱ ❍❛♥♦✐ P❡❞❛❣♦❣✐❝❛❧ ❯♥✐✈❡rs✐t② ✷✱ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s
❛♥❞ t❤❡ t❡❛❝❤❡rs ✐♥ t❤❡ ❛♥❛❧②s✐s ❣r♦✉♣ ❛s ✇❡❧❧ ❛s t❤❡ t❡❛❝❤❡rs ✐♥✈♦❧✈❡❞✳
❚❤❡ ❧❡❝t✉r❡rs ❤❛✈❡ ✐♠♣❛rt❡❞ ✈❛❧✉❛❜❧❡ ❦♥♦✇❧❡❞❣❡ ❛♥❞ ❢❛❝✐❧✐t❛t❡❞ ❢♦r ♠❡
t♦ ❝♦♠♣❧❡t❡ t❤❡ ❝♦✉rs❡ ❛♥❞ t❤❡ t❤❡s✐s✳
■♥ ♣❛rt✐❝✉❧❛r✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② ❞❡❡♣ r❡s♣❡❝t ❛♥❞ ❣r❛t✐t✉❞❡
t♦ ❉r✳ ❉♦ ❍♦❛♥❣ ❙♦♥✱ ✇❤♦ ❤❛s ❞✐r❡❝t ❣✉✐❞❛♥❝❡✱ ❤❡❧♣ ♠❡ ❝♦♠♣❧❡t❡ t❤✐s
t❤❡s✐s✳
❍❛♥♦✐✱ ▼❛② ✻✱ ✷✵✶✾
❙t✉❞❡♥t

❉♦❛♥ ❚❤✐ P❤✉♦♥❣


❈♦♥❢✐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✳ ■ ❛❧s♦ ❛ss✉r❡ t❤❛t t❤❡ ✉s❡❞ ❧✐t❡r❛t✉r❡
❛♥❞ t❤❡ ♦t❤❡r ❛✉①✐❧✐❛r② r❡s♦✉r❝❡s ❤❛✈❡ ❜❡❡♥ ❝♦♠♣❧❡t❡❧② r❡❢❡r❡♥❝❡❞✳
❍❛♥♦✐✱ ▼❛② ✻✱ ✷✵✶✾
❙t✉❞❡♥t

❉♦❛♥ ❚❤✐ P❤✉♦♥❣


❈♦♥t❡♥ts
❚❛❜❧❡ ♦❢ ◆♦t❛t✐♦♥s


✐✐✐

Pr❡❢❛❝❡

✶

✶ ❇❛❝❦❣r♦✉♥❞ ✐♥ ♠❡❛s✉r❡ t❤❡♦r②

✷

✶✳✶✳ ▼❡❛s✉r❡ ♦♥ ❛ s❡t ❛❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷✳ ❉✐❢❢✉s❡ ♠❡❛s✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷✳✶✳ ❖✉t❡r ♠❡❛s✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷✳✷✳ ❉✐❢❢✉s❡ t❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✸✳ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺


✶✳✸✳✶✳ ▲❡❜❡s❣✉❡ ♦✉t❡r ♠❡❛s✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✸✳✷✳ ▼❡❛s✉r❛❜❧❡ s❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✸✳✸✳ ❙❡ts ♦❢ ♠❡❛s✉r❡ ③❡r♦ ❛♥❞ ❝♦♠♣❧❡t❡♥❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸✳✹✳ ❚r❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸✳✺✳ ❇♦r❡❧ s❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✸✳✻✳ ❇♦r❡❧ r❡❣✉❧❛r✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✷ ❚❤❡ ❞❡❢✐♥✐t✐♦♥ ❛♥❞ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s
❛♥❞ ❞✐♠❡♥s✐♦♥
✶✷

✐



✷✳✶✳ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✶✳✶✳ ❉❡❢✐♥✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✶✳✷✳ ❙♦♠❡ ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✷✳ ❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷✳✶✳ ❉❡❢✐♥✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷✳✷✳ ❙♦♠❡ ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸ ❙♦♠❡ ❡①❛♠♣❧❡s

✷✹

✸✳✶✳ ❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥ ♦❢ ❈❛♥t♦r s❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✶✳✶✳ ❈❛♥t♦r s❡ts ✐♥ R1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✶✳✷✳ ●❡♥❡r❛❧✐③❡❞ ❈❛♥t♦r s❡ts ✐♥ R1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✳✸✳ ❈❛♥t♦r s❡ts ✐♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✷✳ ❲❡✐❡rstr❛ss ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✸✳ ❙✐❡r♣✐♥s❦✐ tr✐❛♥❣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✹✳ ❋✐❜♦♥❛❝❝✐ ✇♦r❞ ❢r❛❝t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

❈♦♥❝❧✉s✐♦♥s

✸✹

❘❡❢❡r❡♥❝❡s

✸✺

✐✐



❚❛❜❧❡ ♦❢ ◆♦t❛t✐♦♥s
R

t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✳

Rn

t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳

d(E)

sup {d(x, y)|x ∈ E, y ∈ E}✳

E

t❤❡ ❝❧♦s✉r❡ ❤✉❧❧ ♦❢ ❛ s❡t ❊✳

x
B(x, r)

t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ ❛ ✈❡❝t♦r x✳

{y : d(x, y) < r}✱ t❤❡ ♦♣❡♥ ❜❛❧❧ ♦❢ ❝❡♥t❡r x ❛♥❞ ♦❢ r❛✲
❞✐✉s r ✐♥ Rn ✳

Ln

t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ Rn ✳


A+a

{x + a : x ∈ A}✳

❝❛r❞ A

t❤❡ ♥✉♠❜❡r ♣♦✐♥ts ✐♥ t❤❡ s❡t ❆❀ ♣♦ss✐❜❧② 0 ♦r ∞✳

α(n)

Ln {x ∈ Rn : |x| ≤ 1}✱ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ✉♥✐t ❜❛❧❧✳

✐✐✐


Pr❡❢❛❝❡
❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥✱ ❢✐rst ✐♥tr♦❞✉❝❡❞ ❜② ❋❡❧✐① ❍❛✉s❞♦r❢❢ ✐♥ ✶✾✶✽✱ ✐s
❛♥ ❡①t❡♥❞❡❞ ❝♦♥❝❡♣t ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♥❝❡♣t ♦❢ r❡❛❧ s♣❛❝❡✳ ■t ✐s ❝❤❛r✲
❛❝t❡r✐③❡❞ ❜② ❛ s♣❡❝✐❛❧ ♠❡❛s✉r❛❜❧❡ ❝❧❛ss✱ ❝❛❧❧❡❞ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡✳ ❚❤❡
❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❛♥❞ t❤❡ ❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥ ❜❡❝♦♠❡ ❛♥ ✐♠♣♦rt❛♥t
t♦♦❧ ✐♥ t❤❡ st✉❞② ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ♦❢ s❡ts✳
❚❤❡ ❛✐♠ ♦❢ t❤✐s t❤❡s✐s ✐s t♦ ♣r❡s❡♥t t❤❡ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❛♥❞
❞✐♠❡♥s✐♦♥ ♦♥ Rn ✳ ❇❡s✐❞❡s✱ ✇❡ ❛❧s♦ ♣r♦✈✐❞❡ s♦♠❡ ❡①❛♠♣❧❡s ❛s ❛♣♣❧✐❝❛t✐♦♥s
♦❢ t❤❡ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❛♥❞ ❞✐♠❡♥s✐♦♥✳
❚❤❡ t❤❡s✐s ❝♦♥s✐sts ♦❢ t❤r❡❡ ❝❤❛♣t❡rs ❛s ❢♦❧❧♦✇s✳
■♥ ❈❤❛♣t❡r ✶✱ ✇❡ ✇✐❧❧ r❡✈✐❡✇ ♠❡❛s✉r❡ t❤❡♦r② ♥❡❡❞❡❞ ❢♦r t❤❡ r❡s✉❧ts
♦❢ t❤✐s t❤❡s✐s✳
■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ♣r❡s❡♥t t❤❡ ❞❡❢✐♥✐t✐♦♥ ❛♥❞ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢
❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❛♥❞ ❞✐♠❡♥s✐♦♥✳ ❲❡ ❜❡❣✐♥ ✇✐t❤ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡✳

❚❤❡♥ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥✳ ◆❡①t✱ ✇❡ s❤♦✇
s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥✱ ❡s♣❡❝✐❛❧❧② t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡

n✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✐s ❡q✉❛❧ t♦ n✳
■♥ ❈❤❛♣t❡r ✸✱ ✇❡ ❣✐✈❡ s♦♠❡ ❡①❛♠♣❧❡s ♦❢ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❛♥❞
❞✐♠❡♥s✐♦♥✳

✶


❈❤❛♣t❡r ✶
❇❛❝❦❣r♦✉♥❞ ✐♥ ♠❡❛s✉r❡ t❤❡♦r②
■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ r❡✈✐❡✇ s♦♠❡ r❡s✉❧ts ✐♥ ♠❡❛s✉r❡ t❤❡♦r②✳ ❚❤❡
❝♦♥t❡♥t ♦❢ t❤✐s ❝❤❛♣t❡r ✐s ❜❛s❡❞ ♦♥ ❬✶✱ ✻❪✳

✶✳✶✳

▼❡❛s✉r❡ ♦♥ ❛ s❡t ❛❧❣❡❜r❛

❉❡❢✐♥✐t✐♦♥ ✶✳✶✳ ▲❡t ❳ ❜❡ ❛ s❡t✳ ❆ ❢❛♠✐❧② M ♦❢ s✉❜s❡ts ♦❢ ❳ ✐s ❝❛❧❧❡❞ ❛♥
❛❧❣❡❜r❛ ✐❢✿

a) ∅ ∈ M

❛♥❞ ❳

∈ M✱

b)


■❢

A ∈ M✱

c)

■❢

A1 , A2 , · · ·, An ∈ M✱

t❤❡♥ ❳

\

❆

∈ M✱

n

Ai ∈ M✳

t❤❡♥

i=1

❉❡❢✐♥✐t✐♦♥ ✶✳✷✳
σ ✲❛❧❣❡❜r❛
a) ∅ ∈ M


▲❡t ❳ ❜❡ ❛ s❡t✳ ❆ ❢❛♠✐❧②

M

♦❢ s✉❜s❡ts ♦❢ ❳ ✐s ❝❛❧❧❡❞ ❛

✐❢✿
❛♥❞ ❳

∈ M✱

b)

■❢

A ∈ M✱

c)

■❢

A1 , A2 , · · · ∈ M✱

t❤❡♥ ❳

\

❆

∈ M✱

∞

Ai ∈ M✳

t❤❡♥

i=1

❉❡❢✐♥✐t✐♦♥ ✶✳✸✳ ▲❡t M ❜❡ ❛♥ ❛❧❣❡❜r❛✳ ❆ s❡t ❢✉♥❝t✐♦♥ µ : M → [0, ∞] =
{t : 0 ≤ t ≤ ∞}

✐s ❝❛❧❧❡❞ ❛ ♠❡❛s✉r❡ ✐❢✿

a) µ(∅) = 0✱
b) µ(A) ≤ µ(B)

✇❤❡♥❡✈❡r

A⊂B

❛♥❞

✷

A, B ∈ M,


∞

∞


c) µ

≤

Ai
i=1

µ(Ai )

A1 , A2 , · · · ∈ M✳

✇❤❡♥❡✈❡r

i=1

❉❡❢✐♥✐t✐♦♥ ✶✳✹✳
+∞; σ ✲❢✐♥✐t❡

▲❡t

M

µ

❜❡ ❛♥ ❛❧❣❡❜r❛✳ ❆ ♠❡❛s✉r❡

✐s ❢✐♥✐t❡ ✐❢

µ(X) <


✐❢

X = ∪∞
i=1 Xi ; Xi ∈ M, µ(Xi ) < +∞.

❉❡❢✐♥✐t✐♦♥ ✶✳✺✳

❆ s❡t

A⊂X

✐s

µ

♠❡❛s✉r❛❜❧❡ ✐❢

µ(E) = µ(E ∩ A) + µ(E \ A)

❢♦r ❛❧❧

E ⊂ X.

❲❡ s❤❛❧❧ ❣✐✈❡ ❛ ❢❡✇ s✐♠♣❧❡ ❡①❛♠♣❧❡s✳

❊①❛♠♣❧❡ ✶✳✶✳ 1) M
♠❡♥ts ♦❢

2) M


✐s ❛♥ ❛❧❣❡❜r❛ ❛♥❞

µ(A)

❡q✉❛❧s t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡✲

A✳

✐s ❛♥ ❛❧❣❡❜r❛✳ ▲❡t

x0

❜❡ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t ♦❢

µ(A) =

1

✐❢

x0 ∈ A,

0

✐❢

x0 ∈
/ A.


X

❛♥❞

∀A ∈ M✿

❲❡ ❝♦❧❧❡❝t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ♠❡❛s✉r❛❜❧❡ s❡ts ✐♥
t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳

❚❤❡♦r❡♠ ✶✳✶✳
µ

▲❡t

µ

❜❡ ❛ ♠❡❛s✉r❡ ♦♥

■❢

µ(A) = 0✱

(2)

■❢

A1 , A2 , · · · ∈ C

t❤❡♥


A ∈ C.
❛r❡ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t✱ t❤❡♥

∞

µ

∞

Ai

=

i=1

(3)

■❢ ■❢

❛♥❞ ❧❡t

X✳

♠❡❛s✉r❛❜❧❡ s✉❜s❡ts ♦❢

(1)

X

A1 , A2 , · · · ∈ C ✱


µ (Ai ).
i=1

t❤❡♥

∞

i) µ

Ai
i=1

= lim µ (Ai )
i→∞

♣r♦✈✐❞❡❞

A1 ⊂ A2 ⊂ ...✱

✸

C

❜❡ t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧


∞

ii) µ


Ai

= lim µ (Ai )

i=1

✶✳✷✳

♣r♦✈✐❞❡❞

i→∞

A1 ⊃ A2 ⊃ ...

❛♥❞

µ(A1 ) < ∞.

❉✐❢❢✉s❡ ♠❡❛s✉r❡

✶✳✷✳✶✳

❖✉t❡r ♠❡❛s✉r❡

▲❡t ❛ s❡t ❢✉♥❝t✐♦♥ µ∗ ❜❡ ❞❡❢✐♥❡❞ ♦♥ ❝❧❛ss ♦❢ ❛❧❧ s✉❜s❡ts ♦❢ X ✳ ❚❤❡
s❡t ❢✉♥❝t✐♦♥ µ∗ ✐s ❝❛❧❧❡❞ ❛♥ ♦✉t❡r
a) µ∗ (A) ≥ 0
∀A ⊂ X.


♠❡❛s✉r❡

✐❢

b) µ∗ (∅) = 0 ;
c) A ⊂

∪∞
i=1 Ai

∞

∗

µ∗ (Ai ).

⇒ µ (A) ≤
i=1

◆♦t❛t✐♦♥✿ ❋r♦♠ ❝✮✱ ✇❡ ✐♥❢❡r t❤❛t A ⊂ B ⇒ µ∗ (A) ≤ µ∗ (B).
❚❤❡♦r❡♠ ✶✳✷ ✭❈❛r❛t❤➨♦❞♦r②✮✳ ▲❡t µ∗

❜❡ ❛♥ ♦✉t❡r ♠❡❛s✉r❡ ♦♥ ❳ ❛♥❞

L

❜❡ ❝❧❛ss ♦❢ ❛❧❧ s✉❜s❡ts ❆ ♦❢ ❳ s✉❝❤ t❤❛t

µ∗ (E) = µ∗ (E ∩ A) + µ∗ (E\A)
L


✐s ❛

σ ✲❛❧❣❡❜r❛

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥

❚❤❡ s❡t ❆ s❛t✐s❢②✐♥❣

✶✳✷✳✷✳

(1.1)

µ = µ∗ /L

✐s ❝❛❧❧❡❞

µ∗

✭✶✳✶✮

∀E ⊂ X.

✐s ❛ ♠❡❛s✉r❡ ♦✈❡r

L.

✲ ♠❡❛s✉r❛❜❧❡✳

❉✐❢❢✉s❡ t❤❡♦r❡♠


❚❤❡♦r❡♠ ✶✳✸✳
❋♦r ❡❛❝❤

▲❡t ♠ ❜❡ ❛ ♠❡❛s✉r❡ ♦✈❡r ❛♥ ❛❧❣❡❜r❛

C✳

A ⊂ X✿

∞
∗

m(Pi ) : ∪∞
i=1 Pi ⊃ A; Pi ∈ C

µ (A) = inf

,

✭✶✳✷✮

i=1
t❤❡♥

µ∗

✐s ❛♥ ♦✉t❡r ♠❡❛s✉r❡ ❛♥❞

❜❡❧♦♥❣s t♦


σ ✲❛❧❣❡❜r❛ F(C)

✐s

µ∗

µ∗ (A) = m(A) ∀A ∈ C

✲ ♠❡❛s✉r❛❜❧❡✳

✹

❛♥❞ ❡✈❡r② s❡t


❚❤❡♦r❡♠ ✶✳✹✳ ▲❡t ♠ ❜❡ ❛ ♠❡❛s✉r❡ ♦✈❡r ❛♥ ❛❧❣❡❜r❛ C ✳ ❚❤❡r❡ ✐s ❛ ♠❡❛s✉r❡
σ ✲❛❧❣❡❜r❛ L ⊃ F(C) ⊃ C

µ

♦♥

✐✮

µ(A) = m(A)

✐✐✮
✐✐✐✮


µ

s✉❝❤ t❤❛t✿

∀A ∈ C.

σ

σ

✐s ❢✐♥✐t❡ ✭ ✲❢✐♥✐t❡✮ ✐❢ ♠ ✐s ❢✐♥✐t❡ ✭ ✲❢✐♥✐t❡✮✳

µ

✐s ❢✉❧❧ ♠❡❛s✉r❡✳

✐✈✮ ❆ s❡t ❆ ❜❡❧♦♥❣s t♦ ❢❛♠✐❧②

L

✐❢ ❛♥❞ ♦♥❧② ✐❢ ❆ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡

❢♦r♠✿

A=B\N

♦r

A = B ∪ N,


B ∈ F(C), N ⊂ E ∈ F(C), µ∗ (E) = µ(E) = 0

✇❤❡r❡

♠❡❛s✉r❡ t❤❛t ✐s ❞❡❢✐♥❡❞ ❢r♦♠ ♠ ❜② t❤❡ ❢♦r♠✉❧❛

✶✳✸✳

▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥

✶✳✸✳✶✳

❛♥❞

µ∗

✐s ♦✉t❡r

(1.2)✳

Rn

▲❡❜❡s❣✉❡ ♦✉t❡r ♠❡❛s✉r❡

❲❡ ✉s❡ r❡❝t❛♥❣❧❡s ❛s ♦✉r ❡❧❡♠❡♥t❛r② s❡ts✱ ❞❡❢✐♥❡❞ ❛s ❢♦❧❧♦✇s✳

❉❡❢✐♥✐t✐♦♥ ✶✳✻✳

❆♥


n✲❞✐♠❡♥s✐♦♥❛❧✱❝❧♦s❡❞

r❡❝t❛♥❣❧❡ ✇✐t❤ s✐❞❡s ♦r✐❡♥t❡❞

♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✱ ♦r r❡❝t❛♥❣❧❡ ❢♦r s❤♦rt✱ ✐s ❛ s✉❜s❡t

R ⊂ Rn

♦❢ t❤❡ ❢♦r♠

R = [a1 , b1 ] × [a2 , b2 ] × ... × [an , bn ] ,
✇❤❡r❡

−∞ < ai ≤ bi < ∞

❢♦r

i = 1, ..., n.

❚❤❡ ✈♦❧✉♠❡

µ(R)

♦❢

R

✐s

µ(R) = (b1 − a1 ) (b2 − a2 ) ... (bn − an ) .

■❢

n = 1

♦r

n = 2✱

t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ r❡❝t❛♥❣❧❡ ✐s ✐ts ❧❡♥❣t❤ ♦r

❛r❡❛✱ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ ❡♠♣t② s❡t t♦ ❜❡ ❛ r❡❝t❛♥❣❧❡ ✇✐t❤

µ(∅) = 0✳
R(Rn )✱

♦r

❲❡ ❞❡♥♦t❡ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧

R

✇❤❡♥

n

n✲❞✐♠❡♥s✐♦♥❛❧

✐s ✉♥❞❡rst♦♦❞✱ ❛♥❞ t❤❡♥

R → µ(R)


µ : R (Rn ) → [0, ∞) .
✺

r❡❝t❛♥❣❧❡s ❜②

❞❡❢✐♥❡s ❛ ♠❛♣


❚❤❡ ✉s❡ ♦❢ t❤✐s ♣❛rt✐❝✉❧❛r ❝❧❛ss ♦❢ ❡❧❡♠❡♥t❛r② s❡ts ✐s ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✳ ❲❡
❝♦✉❧❞ ❡q✉❛❧❧② ✇❡❧❧ ✉s❡ ♦♣❡♥ ♦r ❤❛❧❢✲♦♣❡♥ r❡❝t❛♥❣❧❡s✱ ❝✉❜❡s✱ ❜❛❧❧s✱ ♦r ♦t❤❡r
s✉✐t❛❜❧❡ ❡❧❡♠❡♥t❛r② s❡ts❀ t❤❡ r❡s✉❧t ✇♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳

❉❡❢✐♥✐t✐♦♥ ✶✳✼✳ ❚❤❡ ♦✉t❡r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ µ∗ (E) ♦❢ ❛ s✉❜s❡ts E ⊂ Rn ✱
♦r ♦✉t❡r ♠❡❛s✉r❡ ❢♦r s❤♦rt✱ ✐s

∞
∗

∞

µ(Ri ) : E ⊂

µ (E) = inf

i=1

i=1

Ri , Ri ∈ R(Rn ) ,


✇❤❡r❡ t❤❡ ✐♥❢✐♠✉♠ ✐s t❛❦❡♥ ♦✈❡r ❛❧❧ ❝♦✉♥t❛❜❧❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ r❡❝t❛♥❣❧❡s
✇❤♦s❡ ✉♥✐♦♥ ❝♦♥t❛✐♥s

E✳

❚❤❡ ♠❛♣

µ∗ : P(Rn ) → [0, ∞] ,

µ∗ : E → µ∗ (E),

✐s ❝❛❧❧❡❞ ♦✉t❡r ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

∞

µ(Ri )

■♥ t❤✐s ❞❡❢✐♥✐t✐♦♥✱ ❛ s✉♠

µ∗ (E)

❛♥❞

♠❛② t❛❦❡ t❤❡ ✈❛❧✉❡

∞✳

i=1


❊①❛♠♣❧❡ ✶✳✷✳ ▲❡t E = Q ∩ [0, 1] ❜❡ t❤❡ s❡t ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥
✵ ❛♥❞ ✶✳ ❚❤❡♥

E

❤❛s ♦✉t❡r ♠❡❛s✉r❡ ③❡r♦✳ ❚♦ ♣r♦✈❡ t❤✐s✱ ❧❡t

❜❡ ❛♥ ❡♥✉♠❡r❛t✐♦♥ ♦❢ t❤❡ ♣♦✐♥ts ✐♥

/2i

✇❤✐❝❤ ❝♦♥t❛✐♥s

qi ✳

❚❤❡♥

E⊂

E✳

●✐✈❡♥

> 0✱

❧❡t

Ri

{qi : i ∈ N}


❜❡ ❛♥ ✐♥t❡r✈❛❧

∞
i=1 µ(Ri ) s♦
∞

∗

0 ≤ µ (E) ≤

µ(Ri ) = .
i=1

❍❡♥❝❡

µ∗ (E) = 0

s✐♥❝❡

> 0

✐s ❛r❜✐tr❛r②✳ ❚❤❡ s❛♠❡ ❛r❣✉♠❡♥t s❤♦✇s

t❤❛t ❛♥② ❝♦✉♥t❛❜❧❡ s❡t ❤❛s ♦✉t❡r ♠❡❛s✉r❡ ③❡r♦✳ ◆♦t❡ t❤❛t ✐❢ ✇❡ ❝♦✈❡r

E

❜② ❛ ❢✐♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✐♥t❡r✈❛❧s✱ t❤❡♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧s ✇♦✉❧❞
❤❛✈❡ t♦ ❝♦♥t❛✐♥


[0, 1]

s✐♥❝❡

E

✐s ❞❡♥s❡ ✐♥

❧❡❛st ♦♥❡✳

✻

[0, 1]

s♦ t❤❡✐r ❧❡♥❣t❤s s✉♠ t♦ ❛t


❚❤❡♦r❡♠ ✶✳✺✳

▲❡❜❡s❣✉❡ ♦✉t❡r ♠❡❛s✉r❡

µ∗

❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✳

(a) µ∗ (∅) = 0;
(b)

✐❢


E ⊂ F, t❤❡♥ µ∗ (E) ≤ µ∗ (F );

(c)

✐❢

{Ei ⊂ Rn : i ∈ N } ✐s

❛ ❝♦✉♥t❛❜❧❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ s✉❜s❡ts ♦❢

∞
∗

µ

t❤❡♥

∞

Ei

µ∗ (Ei ).

≤

i=1

✶✳✸✳✷✳


Rn ✱

i=1

▼❡❛s✉r❛❜❧❡ s❡ts

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❈❛r❛t❤➨♦❞♦r② ❞❡❢✐♥✐t✐♦♥ ♦❢ ♠❡❛s✉r❛❜✐❧✐t②✳

❉❡❢✐♥✐t✐♦♥ ✶✳✽✳

❆ s✉❜s❡t

A ⊂ Rn

✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ ✐❢

µ∗ (E) = µ∗ (E ∩ A) + µ∗ (E ∩ Ac ),
❢♦r ❡✈❡r② s✉❜s❡t
s❡ts ✐♥

Rn

❜②

E ⊂ Rn ✳ ❲❡ ❞❡♥♦t❡ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡

L(Rn )✳

❚❤❡♦r❡♠ ✶✳✻✳
σ ✲❛❧❣❡❜r❛

L(Rn )

♦♥

❚❤❡ ❝♦❧❧❡❝t✐♦♥

Rn ✱

♦❢ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ s❡ts ✐s ❛

❛♥❞ t❤❡ r❡str✐❝t✐♦♥ ♦❢ ▲❡❜❡s❣✉❡ ♦✉t❡r ♠❡❛s✉r❡

✐s ❛ ♠❡❛s✉r❡ ♦♥

❉❡❢✐♥✐t✐♦♥ ✶✳✾✳

L(Rn )

t♦

L(Rn )✳

▲❡❜❡s❣✉❡ ♠❡❛s✉r❡

µ : L(Rn ) → [0, ∞] ,

µ = µ∗ |L(Rn )

✐s t❤❡ r❡str✐❝t✐♦♥ ♦❢ ▲❡❜❡s❣✉❡ ♦✉t❡r ♠❡❛s✉r❡
❛❜❧❡ s❡ts


µ∗

µ∗

t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r✲

L(Rn )✳

Pr♦♣♦s✐t✐♦♥ ✶✳✶✳

❊✈❡r② r❡❝t❛♥❣❧❡ ✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡✳

✼


✶✳✸✳✸✳

❙❡ts ♦❢ ♠❡❛s✉r❡ ③❡r♦ ❛♥❞ ❝♦♠♣❧❡t❡♥❡ss

Pr♦♣♦s✐t✐♦♥ ✶✳✷✳

■❢

N ⊂ Rn

❛♥❞

s✉r❛❜❧❡✱ ❛♥❞ t❤❡ ♠❡❛s✉r❡ s♣❛❝❡


❉❡❢✐♥✐t✐♦♥ ✶✳✶✵✳
>0

❡✈❡r②

❆ s✉❜s❡t

µ∗ (N ) = 0✱

(Rn , L(Rn ), µ)

N ⊂ Rn

N

✐s ▲❡❜❡s❣✉❡ ♠❡❛✲

✐s ❝♦♠♣❧❡t❡✳

❤❛s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ③❡r♦ ✐❢ ❢♦r

t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❡❝t❛♥❣❧❡s

s✉❝❤ t❤❛t

∞

Ri ,

µ(Ri ) < .


i=1

❊①❛♠♣❧❡ ✶✳✸✳

{Ri : i ∈ N}

∞

N⊂

t❤✐r❞s✬ ❢r♦♠

t❤❡♥

i=1

❚❤❡ st❛♥❞❛r❞ ❈❛♥t♦r s❡t✱ ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ✬♠✐❞❞❧❡

[0, 1]✱ ✐s ❛♥ ✉♥❝♦✉♥t❛❜❧❡ s❡t ♦❢ ③❡r♦ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡

♠❡❛s✉r❡✳

❊①❛♠♣❧❡ ✶✳✹✳

❚❤❡

x✲❛①✐s

✐♥


R2

A = (x, 0) ∈ R2 : x ∈ R ,
❤❛s ③❡r♦ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

✶✳✸✳✹✳

❚r❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡

❆♥ ✐♠♣♦rt❛♥t ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt② ♦❢ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ✐s ✐ts tr❛♥s✲
❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡✳ ■❢ A ⊂ Rn ❛♥❞ h ∈ Rn ✱ ❧❡t

A + h = {x + h : x ∈ A} ,
❞❡♥♦t❡ t❤❡ tr❛♥s❧❛t✐♦♥ ♦❢ A ❜② h✳

Pr♦♣♦s✐t✐♦♥ ✶✳✸✳

■❢

A ⊂ Rn

❛♥❞

h ∈ Rn ✱

t❤❡♥

µ∗ (A + h) = µ∗ (A),
❛♥❞


A+h

✐s ♠❡❛s✉r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢

✽

A

✐s ♠❡❛s✉r❛❜❧❡✳


✶✳✸✳✺✳

❇♦r❡❧ s❡ts

❚❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ♠❡❛s✉r❡ ❛♥❞ t♦♣♦❧♦❣② ✐s ♥♦t ❛ s✐♠♣❧❡
♦♥❡✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ s❤♦✇ t❤❛t ❛❧❧ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞ s❡ts ✐♥ Rn ✱ ❛♥❞
t❤❡r❡❢♦r❡ ❛❧❧ ❇♦r❡❧ s❡ts ✭✐✳❡✳ s❡ts t❤❛t ❜❡❧♦♥❣ t♦ t❤❡ σ ✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞
❜② t❤❡ ♦♣❡♥ s❡ts✮✱ ❛r❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡✳
▲❡t T (Rn ) ⊂ P(Rn ) ❞❡♥♦t❡ t❤❡ st❛♥❞❛r❞ t♦♣♦❧♦❣② ♦♥ Rn ❝♦♥s✐st✐♥❣
♦❢ ❛❧❧ ♦♣❡♥ s❡ts✳ ❚❤❛t ✐s✱ G ⊂ Rn ❜❡❧♦♥❣s t♦ T (Rn ) ✐❢ ❢♦r ❡✈❡r② x ∈ G
t❤❡r❡ ❡①✐sts r > 0 s✉❝❤ t❤❛t Br (x) ⊂ G✱ ✇❤❡r❡

Br (x) = {y ∈ Rn : |x − y| < r} ,
✐s t❤❡ ♦♣❡♥ ❜❛❧❧ ♦❢ r❛❞✐✉s r ❝❡♥t❡r❡❞ ❛t x ∈ Rn ✳

❉❡❢✐♥✐t✐♦♥ ✶✳✶✶✳

❚❤❡ ❇♦r❡❧


❣❡♥❡r❛t❡❞ ❜② t❤❡ ♦♣❡♥ s❡ts✱
❇♦r❡❧

σ ✲❛❧❣❡❜r❛
❙✐♥❝❡

σ ✲❛❧❣❡❜r❛ B(Rn )

B(Rn ) = σ(T (Rn ))✳

σ ✲❛❧❣❡❜r❛

σ ✲❝♦♠♣❛❝t

Rn

✐s t❤❡

σ ✲❛❧❣❡❜r❛

❆ s❡t t❤❛t ❜❡❧♦♥❣s t♦ t❤❡

✐s ❝❛❧❧❡❞ ❛ ❇♦r❡❧ s❡t✳
❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥✱ t❤❡ ❇♦r❡❧

❛❧❣❡❜r❛ ✐s ❛❧s♦ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❝❧♦s❡❞ s❡ts ✐♥
✐s

♦♥


Rn ✳

▼♦r❡♦✈❡r✱ s✐♥❝❡

✭✐✳❡✳ ✐t ✐s ❛ ❝♦✉♥t❛❜❧❡ ✉♥✐♦♥ ♦❢ ❝♦♠♣❛❝t s❡ts✮ ✐ts ❇♦r❡❧

σ✲
Rn
σ✲

❛❧❣❡❜r❛ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❝♦♠♣❛❝t s❡ts✳

Pr♦♣♦s✐t✐♦♥ ✶✳✹✳

❊✈❡r② ♦♣❡♥ s❡t ✐♥

Rn

✐s ❛ ❝♦✉♥t❛❜❧❡ ✉♥✐♦♥ ♦❢ ❛❧♠♦st

❞✐s❥♦✐♥t r❡❝t❛♥❣❧❡s✳

Pr♦♣♦s✐t✐♦♥ ✶✳✺✳ ❚❤❡ ❇♦r❡❧ ❛❧❣❡❜r❛ B(Rn ) ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❝♦❧❧❡❝t✐♦♥
♦❢ r❡❝t❛♥❣❧❡s

✶✳✸✳✻✳

R(Rn )✳


❊✈❡r② ❇♦r❡❧ s❡t ✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡✳

❇♦r❡❧ r❡❣✉❧❛r✐t②

❘❡❣✉❧❛r✐t② ♣r♦♣❡rt✐❡s ♦❢ ♠❡❛s✉r❡s r❡❢❡r t♦ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❛♣✲
♣r♦①✐♠❛t✐♥❣ ✐♥ ♠❡❛s✉r❡ ♦♥❡ ❝❧❛ss ♦❢ s❡ts ✭❢♦r ❡①❛♠♣❧❡✱ ♥♦♥ ♠❡❛s✉r❛❜❧❡

✾


s❡ts✮ ❜② ❛♥♦t❤❡r ❝❧❛ss ♦❢ s❡ts ✭❢♦r ❡①❛♠♣❧❡✱ ♠❡❛s✉r❛❜❧❡ s❡ts✮✳ ▲❡❜❡s❣✉❡
♠❡❛s✉r❡ ✐♥ ❇♦r❡❧ r❡❣✉❧❛r ✐♥ t❤❡ s❡♥s❡ t❤❛t ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ s❡ts ❝❛♥
❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✐♥ ♠❡❛s✉r❡ ❢r♦♠ t❤❡ ♦✉ts✐❞❡ ❜② ♦♣❡♥ s❡ts ❛♥❞ ❢r♦♠ t❤❡
✐♥s✐❞❡ ❜② ❝❧♦s❡❞ s❡ts✱ ❛♥❞ t❤❡② ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❇♦r❡❧ s❡ts ✉♣ t♦
s❡ts ♦❢ ♠❡❛s✉r❡ ③❡r♦✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ✐s ❛ s✐♠♣❧❡ ❝r✐t❡r✐♦♥ ❢♦r ▲❡❜❡s❣✉❡
♠❡❛s✉r❛❜✐❧✐t② ✐♥ t❡r♠s ♦❢ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞ s❡ts✳
❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ❡①♣r❡ss❡s ❛ ❢✉♥❞❛♠❡♥t❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣✲
❡rt② ♦❢ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ s❡ts ❜② ♦♣❡♥ ❛♥❞ ❝♦♠♣❛❝t s❡ts✳

❚❤❡♦r❡♠ ✶✳✼✳

■❢

A ⊂ Rn ✱

t❤❡♥

µ∗ (A) = inf {µ(G) : A ⊂ G, G
❛♥❞ ✐❢


A

♦♣❡♥

},

✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡✱ t❤❡♥

µ(A) = sup {µ(K) : K ⊂ A, K

❚❤❡♦r❡♠ ✶✳✽✳
>0

❢♦r ❡✈❡r②

❆ s✉❜s❡t

A ⊂ Rn

❝♦♠♣❛❝t

}.

✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢

t❤❡r❡ ✐s ❛♥ ♦♣❡♥ s❡t

G⊃A

s✉❝❤ t❤❛t


µ∗ (G \ A) < .

❚❤❡♦r❡♠ ✶✳✾✳

❆ s✉❜s❡t

❢♦r ❡✈❡r②

> 0

G⊃A⊃F

❛♥❞

A ⊂ Rn

✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢

t❤❡r❡ ✐s ❛♥ ♦♣❡♥ s❡t

G

❛♥❞ ❛ ❝❧♦s❡❞ s❡t

F

s✉❝❤ t❤❛t

µ(G \ F ) < .

■❢

µ(A) < ∞✱

t❤❡♥

❉❡❢✐♥✐t✐♦♥ ✶✳✶✷✳

F

♠❛② ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ❝♦♠♣❛❝t✳

❚❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡ts ✐♥

s❡❝t✐♦♥s ♦❢ ♦♣❡♥ s❡ts ✐s ❞❡♥♦t❡❞ ❜②

Rn

Gδ (Rn )✱

Rn

t❤❛t ❛r❡ ❝♦✉♥t❛❜❧❡ ✐♥t❡r✲

❛♥❞ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡ts ✐♥

t❤❛t ❛r❡ ❝♦✉♥t❛❜❧❡ ✉♥✐♦♥s ♦❢ ❝❧♦s❡❞ s❡ts ✐s ❞❡♥♦t❡❞ ❜②

Gδ


❛♥❞

Fσ

Fσ (Rn )✳

s❡ts ❛r❡ ❇♦r❡❧✳ ❚❤✉s✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♥❡①t r❡s✉❧t t❤❛t

✶✵


❡✈❡r② ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡ s❡t ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✉♣ t♦ ❛ s❡t ♦❢ ♠❡❛s✉r❡
③❡r♦ ❜② ❛ ❇♦r❡❧ s❡t✳ ❚❤✐s ✐s t❤❡ ❇♦r❡❧ r❡❣✉❧❛r✐t② ♦❢ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

❚❤❡♦r❡♠ ✶✳✶✵✳
t❤❡r❡ ❡①✐sts s❡ts

❙✉♣♣♦s❡ t❤❛t

G ∈ Gδ (Rn )

❛♥❞

G ⊃ A ⊃ F,

❚❤❡♦r❡♠ ✶✳✶✶✳
❇♦r❡❧

A ⊂ Rn


❚❤❡ ▲❡❜❡s❣✉❡

✐s ▲❡❜❡s❣✉❡ ♠❡❛s✉r❛❜❧❡✳ ❚❤❡♥

F ∈ Fσ (Rn )

s✉❝❤ t❤❛t

µ(G \ A) = µ(A \ F ) = 0.
σ ✲❛❧❣❡❜r❛ L(Rn )

σ ✲❛❧❣❡❜r❛ B(Rn )✳

✶✶

✐s t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤❡


❈❤❛♣t❡r ✷
❚❤❡ ❞❡❢✐♥✐t✐♦♥ ❛♥❞ s♦♠❡ ♣r♦♣❡rt✐❡s
♦❢ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❛♥❞
❞✐♠❡♥s✐♦♥
❚❤❡ ❛✐♠ ♦❢ t❤✐s ❝❤❛♣t❡r ✐s t♦ ♣r❡s❡♥t t❤❡ ♥♦t✐♦♥ ♦❢ ❍❛✉s❞♦r❢❢ ♠❡❛✲
s✉r❡s ❛♥❞ ❞✐♠❡♥s✐♦♥✳ ❲❡ ❛❧s♦ ♣r♦✈✐❞❡ s♦♠❡ ♦❢ ✐ts ✐♠♣♦rt❛♥t t❤❡♦r❡♠s
❛♥❞ ❧❡♠♠❛s✳
❚❤❡ ♣r❡s❡♥t❛t✐♦♥ ❣✐✈❡♥ ✐♥ t❤✐s ❝❤❛♣t❡r ❝♦♠❡s ❢r♦♠ t❤❡ r❡s✉❧ts ✐♥ ❬✻❪✳

✷✳✶✳

❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s


✷✳✶✳✶✳

❉❡❢✐♥✐t✐♦♥

▲❡t (X, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡✱ F ❛ ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ X ❛♥❞ ζ ❛
♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ ♦♥ F ✳ ❲❡ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛ss✉♠♣t✐♦♥s✳
✭✶✮ ❋♦r ❡✈❡r② δ > 0 t❤❡r❡ ❛r❡ E1 , E2 , · · · ∈ F s✉❝❤ t❤❛t X =

∞
i=1 Ei

❛♥❞

d(Ei ) ≤ δ.
✭✷✮ ❋♦r ❡✈❡r② δ > 0 t❤❡r❡ ✐s E ∈ F s✉❝❤ t❤❛t ζ(E) ≤ δ ❛♥❞ d(E) ≤ δ.

✶✷


❋♦r 0 < δ ≤ ∞ ❛♥❞ A ⊂ X ✇❡ ❞❡❢✐♥❡
∞

∞

ζ(Ei ) : A ⊂

ψ(A) = lim inf
δ→0


i=1

Ei , d(Ei ) ≤ δ, Ei ∈ F

.

i=1

▲❡t s ∈ [0, ∞) ❛♥❞ ❝❤♦♦s❡

F = {E : E ⊂ X} ,
ζ(E) = ζs (E) = d(E)s ,
✇✐t❤ 00 = 1 ❛♥❞ d(0)s = 0.
❚❤❡♥ t❤❡ ♠❡❛s✉r❡ ψ ✐s ❝❛❧❧❡❞ t❤❡ s✲❞✐♠❡♥s✐♦♥❛❧ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡✱
❞❡♥♦t❡❞ ❜② Hs ✳ ❲❡ ❤❛✈❡

Hs = lim Hδs (A),
δ−→0

✇❤❡r❡ Hδs (A) = inf

d(Ei )s | A ⊂
i

Ei ; d(Ei ) ≤ δ .
i

◆♦t❡✿ ❚❤❡ ✐♥t❡❣r❛❧ ❞✐♠❡♥s✐♦♥❛❧ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ♣❧❛② ❛ s♣❡❝✐❛❧ r♦❧❡✳
✰ ❋♦r s = 0✿ H0 ✐s t❤❡ ❝♦✉♥t✐♥❣ ♠❡❛s✉r❡✳ ■♥❞❡❡❞✱


H0 (A) ❂ ❝❛r❞ ❆ ❂ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ ❆✳
✰ ❋♦r s = 1✿ H1 ✐s ❡q✉❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ ❆ ✐❢ ❆ ✐s ❛ r❡❝t✐❢✐❛❜❧❡ ❝✉r✈❡✳
✰ ❋♦r s = n ✐♥ Rn ✿ Hn = 2n .α(n)−1 .Ln ✳
✰ ❋♦r s > n✿ Hs ✐♥ Rn ✐s ✉♥✐♥t❡r❡st✐♥❣ s✐♥❝❡ Hs (Rn ) = 0.
❲❡ s❤❛❧❧ ♥♦✇ ❞❡r✐✈❡ s♦♠❡ s✐♠♣❧❡ ♣r♦♣❡rt✐❡s ♦❢ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s
✐♥ ❛ ❣❡♥❡r❛❧ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ❳✳

✷✳✶✳✷✳

❙♦♠❡ ♣r♦♣❡rt✐❡s

❚❤❡♦r❡♠ ✷✳✶✳ ▲❡t (X, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡✱ s ∈ [0, n) ❛♥❞ ζ(E) = d(E)s
❢♦r ❊

⊂

❳✳ ■❢

✶✸


(1) F1 = {F ⊂ X : ❋

✐s ❝❧♦s❡❞

(2) F2 = {U ⊂ X : ❯

✐s ♦♣❡♥

(3) X = Rn

t❤❡♥

❛♥❞

}

}

♦r

F3 = {K ⊂ Rn : ❑

ψ(Fi , ζ) = Hs

Pr♦♦❢✳

♦r

✐s ❝♦♥✈❡①

},

(i = 1, 2, 3).

✭✶✮ ❚❛❦❡ ❛♥② A ⊂ X ✳ ❋♦r ❛❧❧ δ > 0✱ ✇❡ ❤❛✈❡

Hδs (A) = inf

d(Ei )s | A ⊂


Ei ; d(Ei ) ≤ δ; Ei ∈ F
i

i

d(Ei )s | A ⊂

≤ inf

Ei ; d(Ei ) ≤ δ; Ei ∈ F1

i

.

i

❍❡♥❝❡✱ ✇❡ ❤❛✈❡

Hs (A) = lim Hδs (A)
δ−→0

d(Ei )s | A ⊂

≤ lim inf
δ−→0

i

Ei ; d(Ei ) ≤ δ; Ei ∈ F1

i

✭✷✳✶✮

= ψ(F1 , ζ)(A).

❙✐♥❝❡ d(Ei ) = d(Ei )✳ ■t ❢♦❧❧♦✇s t❤❛t ✐❢ (Ei ) ✐s δ ✲ ❝♦✈❡r ♦❢ ❆ t❤❡♥ (Ei ) ✐s
❛❧s♦ δ ✲ ❝♦✈❡r ♦❢ ❆✳

Hδs (A) = inf

d(Ei )s | A ⊂
i

i

d(Ei )s | A ⊂

= inf

Ei ; d(Ei ) ≤ δ; Ei ∈ F
i

i

❚❤❡♥✱ Hδs (A) ≥ inf

Ei ; d(Ei ) ≤ δ; Ei ∈ F

d(Ei )s | A ⊂

i

Ei ; d(Ei ) ≤ δ; Ei ∈ F1 .
i

✶✹


❚❤✐s ✐♠♣❧✐❡s✱

Hs (A) = lim Hδs (A)
δ→0

d(Ei )s | A ⊂

≥ lim inf
δ−→0

i

Ei ; d(Ei ) ≤ δ; Ei ∈ F1
i

✭✷✳✷✮

= ψ(F1 , ζ)(A).
❈♦♠❜✐♥✐♥❣ (2.1) ✇✐t❤ (2.2) ②✐❡❧❞s ψ(F1 , ζ)(A) = Hs .
✭✷✮ ❚❛❦❡ ❛♥② A ⊂ X ✳ ❋♦r ❛❧❧ δ > 0✱ ✇❡ ❤❛✈❡

Hδs (A) = inf


d(Ei )s | A ⊂
i

Ei ; d(Ei ) ≤ δ; Ei ∈ F
i

d(Ei )s | A ⊂

≤ inf
i

Ei ; d(Ei ) ≤ δ; Ei ∈ F2

.

i

❍❡♥❝❡✱ ✇❡ ❤❛✈❡

Hs (A) = lim Hδs (A)
δ→0

d(Ei )s | A ⊂

≤ lim inf
δ→0

i


Ei ; d(Ei ) ≤ δ; Ei ∈ F2
i

✭✷✳✸✮

= ψ(F2 , ζ)(A).
■t r❡♠❛✐♥s t♦ ♣r♦✈❡ Hδs (A) ≥ ψ(F2 , ζ)(A).
▲❡t {Ei }∞
i=1 s✉❝❤ t❤❛t d(Ei ) ≤ δ; A ⊂
❛♥❞
i

d(Ei )s < Hδs (A) + δ ✳

∀ε > 0✱ ✇❡ ❞❡♥♦t❡ Eiε = Ei + B(0, ε)✳
❲❡ ❤❛✈❡✿
✰ Eiε ✐s ♦♣❡♥ ∀i✱
✰A⊂
✰

∞

i=1
ε/2i
d(Ei )

ε/2i

Ei


≤ d(Ei ) +

∀ε > 0✱
ε
2i−1 ✳

✶✺

Ei ; Ei ∈ F
i


❚❤❡♥✱ ❢♦r ❡✈❡r② 0 < ε < δ
∞
s

d(Ui ) |d(Ui ) ≤ 2δ, Ui ⊂ F2

inf

ε/2i s

≤

i

d(Ei

)


i=1
∞

≤

d(Ei ) +
i=1

▲❡t ε → 0✱ ✇❡ ♦❜t❛✐♥

∞
i=1

❍❡♥❝❡✱

δ→0

2i−1

s

.

d(Ei )s ≤Hδs (A) + δ ✳

d(Ui )s |d(Ui ) ≤ 2δ, Ui ⊂ F2

ψ(F2 , ζ) = lim inf

ε


i

≤ lim (Hδs (A) + δ)
δ→0

= Hs (A).

✭✷✳✹✮

❋r♦♠ ✭2.3✮ ❛♥❞ ✭2.4✮✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ψ(F2 , ζ) = Hs (A).
✭✸✮ ❋♦r ❛❧❧ A ⊂ Rn ✳ ▲❡t A(c) ❜❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ ❆✳
❲❡ ✇✐❧❧ ♣r♦✈❡ t❤❛t d(A) = d(A(c) )✳ ■♥❞❡❡❞✱
m

A(c) =

k=1

❋♦r ❛❧❧ x =
✇❤❡r❡

m1

m2

λk xk ; y =

k=1


m1

tk = 1; xk ∈ A .

tk xk | m ∈ N, tk ≥ 0, ∀k = 1, m,

λk = 1;

µl y l ∈ A(c) ,

l=1

m2

k=1

k

µl = 1.
l=1

❲❡ ❤❛✈❡✿

d(x, y) = x − y
m2

µl y l

= x−
l=1

m2

µl (x − y l )

=
l=1
m2

≤

µl x − y

m2
l

l

≤ max x − y .

l=1

(s✐♥❝❡

µl = 1)
l=1

✶✻


∀l = 1, m2 ✱ ✇❡ ❤❛✈❡

x − yl =

m1

λk (xk − y l ) ≤

k=1

❚❤✐s ❧❡❛❞s t♦ d(x, y) ≤ d(A)✳

m1

λk xk − y l ≤ max xk − y l ≤ d(A)✳
1,m2

k=1

❙✐♥❝❡ x, y ∈ A(c) ❛r❡ ❛r❜✐tr❛r②✱ ♦♥❡ ❤❛s d(A(c) ) = sup ≤ d(A)✳
x,y∈A(c)

❇✉t A ⊂ A(c) ✱ ✇❡ ♦❜t❛✐♥ d(A) ≤ d(A(c) )✳
❚❤✐s ✐♠♣❧✐❡s t❤❛t d(A) = d(A(c) ) ❢♦r ❛❧❧ A ∈ Rn ✳
❙✐♠✐❧❛r❧② ♣❛rt ✭✶✮✱ ✇❡ ❛❧s♦ ❤❛✈❡ ψ(F3 , ζ) = Hs ✇✐t❤ X = Rn ✳
❚❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡✳

❈♦r♦❧❧❛r② ✷✳✶✳

▲❡t

X = Rn .


❚❤❡♥✱ ❢♦r ❡❛❝❤

s > 0, Hs

✐s ❇♦r❡❧ r❡❣✉❧❛r✳

❖❢t❡♥ ♦♥❡ ✐s ♦♥❧② ✐♥t❡r❡st❡❞ ✐♥ ❦♥♦✇✐♥❣ ✇❤✐❝❤ s❡ts ❤❛✈❡ Hs ♠❡❛s✉r❡
③❡r♦✳ ❋♦r t❤✐s ✐t ✐s ❡♥♦✉❣❤ t♦ ✉s❡ ❛♥② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♠❡❛s✉r❡s Hδs ✱
s
❀ ✐♥ ❢❛❝t ✇❡ ❞♦♥✬t r❡❛❧❧② ♥❡❡❞ ❛♥② ♠❡❛s✉r❡ ❛t ❛❧❧✳ ❲❡
❢♦r ❡①❛♠♣❧❡ H∞

❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳

▲❡♠♠❛ ✷✳✶✳

▲❡t

A ⊂ X, s ∈ [0, ∞)

❛♥❞

δ ∈ (0, ∞).

❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣

❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿

(1) Hs (A) = 0✳

(2) Hδs (A) = 0✳
(3) ∀ε > 0, ∃E1 , E2 , ... ⊂ X

s✉❝❤ t❤❛t

A⊂

Ei
i

Pr♦♦❢✳

d(Ei )s < ε✳

❛♥❞

i

(1) ⇒ (2) : ❖❜✈✐♦✉s❧②

(2) ⇒ (3) : ❲❡ ❤❛✈❡ Hδs (A) = inf

d(Ei )s |A ⊂

Ei ; d(Ei ) ≤ δ ✳
i

i

■t ❢♦❧❧♦✇s t❤❛t ∀ε > 0, ∃E1 , E2 , ... ⊂ X s✉❝❤ t❤❛t✿ A ⊂

❛♥❞
i

i

d(Ei )s − ε < Hδs (A).

❚❤✉s✱

i

Ei ; d(Ei ) ≤ δ

d(Ei )s < ε. ✭s✐♥❝❡ Hδs (A) = 0 ✮

(3) ⇒ (2) :
∀ε > 0, ∃E1 , E2 , ... ⊂ X s✉❝❤ t❤❛t A ⊂

Ei ❛♥❞
i

1
s

❲❡ ♦❜t❛✐♥ ∀i : d(Ei ) < ε ✳
✶✼

d(Ei )s < ε✳
i



❋♦r ❛❧❧ δ > 0✱ ❧❡t ε = δ s ✱ ✇❡ ❤❛✈❡ Hδs (A) = Hs (A) ≤
❚❤✐s ✐♠♣❧✐❡s 0 ≤

lim Hδs (A)
δ→0

≤ lim δ = 0✳
s

d(Ei )s < ε = δ s ✳
i

δ→0

❚❤❡r❡❢♦r❡✱ Hδs (A) = 0✳ ❚❤✐s ❢✐♥✐s❤❡s t❤❡ ♣r♦♦❢✳
❲❡ ✇✐❧❧ ❝♦♠♣❛r❡ ♠❡❛s✉r❡ Hs ✇✐t❤ ❡❛❝❤ ♦t❤❡r✳

❚❤❡♦r❡♠ ✷✳✷✳

▲❡t ❳ ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡✱

(1) Hs (A) < ∞

✐♠♣❧✐❡s

(2) Ht (A) > 0
Pr♦♦❢✳

✐♠♣❧✐❡s


A ⊂ X ✱ 0 ≤ s < t < ∞✳

❚❤❡♥

Ht (A) = 0✱

Hs (A) = ∞✳

(1)✳ ❙✉♣♣♦s❡ t❤❛t Hs (A) < ∞✳ ❚❤❡♥✱ ∀0 < δ < 1✱ ✇❡ ❤❛✈❡

Hδs (A) < ∞✳
❋♦r s = 1✱ ∃A ⊂

Ei ✇✐t❤ d(Ei ) ≤ δ s✉❝❤ t❤❛t
i

d(Ei )s ≤ Hδs (A) + 1 < ∞.
i

■t ✐♠♣❧✐❡s t❤❛t

Hδt (A) ≤

d(Ei )t =
i

d(Ei )s .d(Ei )t−s
i


d(Ei )s .δ t−s ≤ δ t−s .

≤
i

d(Ei )s ≤ δ t−s .(Hδs (A) + 1).
i

❚❤❡♥ Ht (A) = lim Hδt (A) ≤ lim δ t−s (Hδs (A) + 1) = 0✳
δ→0

δ→0

(2). ■❢ Hδs (A) < ∞✱ t❤❡♥ ❜② t❤❡ ♣❛rt (1) ✇❡ ❤❛✈❡ Ht (A) = 0✳ ❲❡
❣❡t t❤❡ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡❞✳
❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ s❛②s t❤❛t ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡s ❜❡❤❛✈❡ ♥✐❝❡❧②
✉♥❞❡r tr❛♥s❧❛t✐♦♥s ❛♥❞ ❞✐❧❛t✐♦♥s ✐♥ Rn ✳

❚❤❡♦r❡♠ ✷✳✸✳

▲❡t

A ⊂ Rn , a ∈ Rn , 0 < t < ∞

✐✮

Hs (A + a) = Hs (A)

✇❤❡r❡


A + a = {x + a : x ∈ A} ,

✐✐✮

Hs (tA) = ts Hs (A)

✇❤❡r❡

tA = {tx : x ∈ A} .
✶✽