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Questions of an example of a measurable function fails to be continuous everywhere or even, almost everywhere

score 660 · Answer 1 · 2026-04-12 23:53:54

660

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Questions of an example of a measurable function fails to be continuous everywhere or even, almost everywhere

Published on 12 Apr 2026 - 23:53

score 304 · Answer 2 · 2026-04-10 21:39:14

304

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Existence of Lebesgue measure on real line proof help

Published on 10 Apr 2026 - 21:39

score 595 · Answer 3 · 2026-04-24 03:52:37

595

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Is $f$ integrable if it is the limit of integrable functions with a uniform bound on their integrals?

Published on 24 Apr 2026 - 3:52

score 119 · Answer 4 · 2026-04-14 00:11:18

119

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Over ZF does "every non-seperable Hilbert space has an orthonormal basis" imply "there exists a non-Lebesgue measurable set"?

Published on 14 Apr 2026 - 0:11

score 789 · Answer 5 · 2026-04-14 11:53:58

789

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The exterior measure of a closed cube is equal to his volume.

Published on 14 Apr 2026 - 11:53

score 3.5k · Answer 6 · 2026-04-20 00:44:45

3.5k

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Proof that outer measure of interval equals length, why use Heine-Borel?

Published on 20 Apr 2026 - 0:44

score 93 · Answer 7 · 2026-04-12 19:24:10

93

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Calculate the measure of a measurable set under nonlinear mapping.

Published on 12 Apr 2026 - 19:24

score 52 · Answer 8 · 2026-04-13 18:10:03

52

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Existence of probability measure, $\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)$

Published on 13 Apr 2026 - 18:10

score 253 · Answer 9 · 2026-04-10 20:04:18

253

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Interpretation of proof of the Lebesgue-Stieltjes measure as a $\sup$ over compact subsets

Published on 10 Apr 2026 - 20:04

score 1.4k · Answer 10 · 2026-04-16 19:44:14

1.4k

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Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.

Published on 16 Apr 2026 - 19:44

score 8.4k · Answer 11 · 2026-04-24 07:05:36

8.4k

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Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Published on 24 Apr 2026 - 7:05

score 3.3k · Answer 12 · 2026-04-15 16:44:11

3.3k

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Lebesgue outer measure of disjoint sets in $\mathbb{R}^n$

Published on 15 Apr 2026 - 16:44

score 1.4k · Answer 13 · 2026-04-16 02:43:33

1.4k

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Prove that jump functions are measurable

Published on 16 Apr 2026 - 2:43

score 104 · Answer 14 · 2026-04-14 03:50:07

104

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Prove $f(x) = \int_{\Bbb {R}} \chi_E (y) \chi_E (y-x)dy$ is continuous, where $E$ is a subset of $\Bbb R$ with finite Lebesgue measure.

Published on 14 Apr 2026 - 3:50

score 545 · Answer 15 · 2026-04-10 22:27:44

545

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counter examples for Riemann and Lebesgue integrabilities

Published on 10 Apr 2026 - 22:27

List Question

Questions of an example of a measurable function fails to be continuous everywhere or even, almost everywhere

Existence of Lebesgue measure on real line proof help

Is $f$ integrable if it is the limit of integrable functions with a uniform bound on their integrals?

Over ZF does "every non-seperable Hilbert space has an orthonormal basis" imply "there exists a non-Lebesgue measurable set"?

The exterior measure of a closed cube is equal to his volume.

Proof that outer measure of interval equals length, why use Heine-Borel?

Calculate the measure of a measurable set under nonlinear mapping.

Existence of probability measure, $\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)$

Interpretation of proof of the Lebesgue-Stieltjes measure as a $\sup$ over compact subsets

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.

Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Lebesgue outer measure of disjoint sets in $\mathbb{R}^n$

Prove that jump functions are measurable

Prove $f(x) = \int_{\Bbb {R}} \chi_E (y) \chi_E (y-x)dy$ is continuous, where $E$ is a subset of $\Bbb R$ with finite Lebesgue measure.

counter examples for Riemann and Lebesgue integrabilities

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