Question

Prove $\lim_{r \to 0} \frac{1}{2r} \int_{x-r}^{x+r} f(z) dz = f(x)$

score 178 · Answer 1 · 2026-03-25 12:49:40

178

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Prove $\lim_{r \to 0} \frac{1}{2r} \int_{x-r}^{x+r} f(z) dz = f(x)$

Published on 25 Mar 2026 - 12:49

score 291 · Answer 2 · 2018-04-08 04:08:00

291

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On a characterization of measure zero sets on $\mathbb{R}$ [Proof Verification]

Published on 08 Apr 2018 - 4:08

score 55 · Answer 3 · 2026-03-25 12:48:50

55

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Integral over product space with one singularity is finite

Published on 25 Mar 2026 - 12:48

score 313 · Answer 4 · 2026-03-25 12:48:37

313

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An counterexample for the monotone convergence theorem and dominated convergence theorem

Published on 25 Mar 2026 - 12:48

score 263 · Answer 5 · 2026-03-24 23:44:22

263

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Lebesgue measure - swap invariance

Published on 24 Mar 2026 - 23:44

score 375 · Answer 6 · 2018-04-09 17:44:12

375

Views

Let $A \subset \mathbb{R}$ be a Lebesgue measurable set with positive Lebesgue measure.

Published on 09 Apr 2018 - 17:44

score 22 · Answer 7 · 2018-04-10 08:58:53

22

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Subset of $\Bbb R$ that doesn’t satisfy $\lim\limits_{\epsilon\rightarrow 0}\mu((E+\epsilon)\backslash E)=0$

Published on 10 Apr 2018 - 8:58

score 115 · Answer 8 · 2026-03-25 07:44:37

115

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What is the definition of $s$-dimensional Lebesgue measure on $R^d$ for $s\ne d$?

Published on 25 Mar 2026 - 7:44

score 124 · Answer 9 · 2026-03-25 14:34:38

124

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Lebesgue integral of non-negative measurable sequence of functions (not monotone)

Published on 25 Mar 2026 - 14:34

score 89 · Answer 10 · 2018-04-11 22:00:55

89

Views

Prove a set has measure 0

Published on 11 Apr 2018 - 22:00

score 362 · Answer 11 · 2026-03-25 14:33:21

362

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Stein-Shakarchi Real ananlysis problem 2.1: a function has integration 0 in any subset of a given interval.

Published on 25 Mar 2026 - 14:33

score 117 · Answer 12 · 2026-03-25 14:33:16

117

Views

Properity about Lebesgue integral with measurable and non-negative functions on measurable and bounded set.

Published on 25 Mar 2026 - 14:33

score 58 · Answer 13 · 2026-03-25 05:37:08

58

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Sectioning sets in product measures

Published on 25 Mar 2026 - 5:37

score 43 · Answer 14 · 2026-03-25 14:29:23

43

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Cauchy functional with local Lebesgue integrability

Published on 25 Mar 2026 - 14:29

score 969 · Answer 15 · 2026-03-25 08:01:49

969

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Lebesgue outer measure of the union is strictly less than the sum of the measures

Published on 25 Mar 2026 - 8:01

List Question

Prove $\lim_{r \to 0} \frac{1}{2r} \int_{x-r}^{x+r} f(z) dz = f(x)$

On a characterization of measure zero sets on $\mathbb{R}$ [Proof Verification]

Integral over product space with one singularity is finite

An counterexample for the monotone convergence theorem and dominated convergence theorem

Lebesgue measure - swap invariance

Let $A \subset \mathbb{R}$ be a Lebesgue measurable set with positive Lebesgue measure.

Subset of $\Bbb R$ that doesn’t satisfy $\lim\limits_{\epsilon\rightarrow 0}\mu((E+\epsilon)\backslash E)=0$

What is the definition of $s$-dimensional Lebesgue measure on $R^d$ for $s\ne d$?

Lebesgue integral of non-negative measurable sequence of functions (not monotone)

Prove a set has measure 0

Stein-Shakarchi Real ananlysis problem 2.1: a function has integration 0 in any subset of a given interval.

Properity about Lebesgue integral with measurable and non-negative functions on measurable and bounded set.

Sectioning sets in product measures

Cauchy functional with local Lebesgue integrability

Lebesgue outer measure of the union is strictly less than the sum of the measures

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