Question

Absolute continuity with respect to Lebesgue measure

score 51 · Answer 1 · 2019-02-07 21:18:23

51

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Absolute continuity with respect to Lebesgue measure

Published on 07 Feb 2019 - 21:18

score 50 · Answer 2 · 2019-02-07 23:11:14

50

Views

Find a Borel set $F$ such that $\lim_{k \to 0} \frac{m_N(F \cap B_k(x))}{m_N(B_k(x))}$ does not exist at some $x$.

Published on 07 Feb 2019 - 23:11

score 41 · Answer 3 · 2026-03-29 04:09:51

41

Views

Show that $\lambda_{F | (F(- \infty), F(\infty)]} = \lambda_{|(F(- \infty), F(\infty)]}$

Published on 29 Mar 2026 - 4:09

score 36 · Answer 4 · 2026-03-25 14:19:50

36

Views

Reference request: Bounded function can be approximated by continuous functions in $L_1$ with bounded $L_\infty$-norm

Published on 25 Mar 2026 - 14:19

score 871 · Answer 5 · 2026-03-25 06:02:17

871

Views

Lebesgue-measurable or Borel-measurable

Published on 25 Mar 2026 - 6:02

score 925 · Answer 6 · 2026-03-25 23:35:50

925

Views

Integral of cantor function

Published on 25 Mar 2026 - 23:35

score 145 · Answer 7 · 2019-02-11 06:49:52

145

Views

Limit $\lim_{n\to \infty} n^2\int _0^1\frac{1}{(1+x^2)^n}$

Published on 11 Feb 2019 - 6:49

score 128 · Answer 8 · 2019-02-11 20:54:04

128

Views

Basic approximation lemma for Lebesgue sets

Published on 11 Feb 2019 - 20:54

score 98 · Answer 9 · 2019-02-13 01:57:26

98

Views

Metric space associated with measure space is path-connected

Published on 13 Feb 2019 - 1:57

score 406 · Answer 10 · 2026-03-27 15:16:53

406

Views

How to prove that Riemann integrable function is measurable?

Published on 27 Mar 2026 - 15:16

score 408 · Answer 11 · 2019-02-13 18:31:58

408

Views

Show m( ∪ ) = () + *() iff there exists measurable sets A1 and B1 such that ⊂ A1 , ⊂ B1 and (A1 ∩ B1 ) = 0.

Published on 13 Feb 2019 - 18:31

score 545 · Answer 12 · 2026-03-25 11:21:48

545

Views

Lebesgue outer measure of image of set is less than or equal to the Lebesgue outer measure of set

Published on 25 Mar 2026 - 11:21

score 120 · Answer 13 · 2026-03-29 05:50:16

120

Views

$ \mu(|f|\geq \alpha) = \frac{1}{\alpha} ||f||_1$

Published on 29 Mar 2026 - 5:50

score 160 · Answer 14 · 2026-03-29 05:43:38

160

Views

Proving that the integral of a function with respect to a measure is finite when the measure is bounded

Published on 29 Mar 2026 - 5:43

score 248 · Answer 15 · 2026-03-29 05:50:22

248

Views

Prove that integral of infinite sum is in $L^1$

Published on 29 Mar 2026 - 5:50

List Question

Absolute continuity with respect to Lebesgue measure

Find a Borel set $F$ such that $\lim_{k \to 0} \frac{m_N(F \cap B_k(x))}{m_N(B_k(x))}$ does not exist at some $x$.

Show that $\lambda_{F | (F(- \infty), F(\infty)]} = \lambda_{|(F(- \infty), F(\infty)]}$

Reference request: Bounded function can be approximated by continuous functions in $L_1$ with bounded $L_\infty$-norm

Lebesgue-measurable or Borel-measurable

Integral of cantor function

Limit $\lim_{n\to \infty} n^2\int _0^1\frac{1}{(1+x^2)^n}$

Basic approximation lemma for Lebesgue sets

Metric space associated with measure space is path-connected

How to prove that Riemann integrable function is measurable?

Show m( ∪ ) = () + *() iff there exists measurable sets A1 and B1 such that ⊂ A1 , ⊂ B1 and (A1 ∩ B1 ) = 0.

Lebesgue outer measure of image of set is less than or equal to the Lebesgue outer measure of set

$ \mu(|f|\geq \alpha) = \frac{1}{\alpha} ||f||_1$

Proving that the integral of a function with respect to a measure is finite when the measure is bounded

Prove that integral of infinite sum is in $L^1$

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