Question

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

score 574 · Answer 1 · 2026-04-15 14:35:07

574

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$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Published on 15 Apr 2026 - 14:35

score 157 · Answer 2 · 2026-04-11 13:40:00

157

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Cauchy formula for repeated Lebesgue integration

Published on 11 Apr 2026 - 13:40

score 205 · Answer 3 · 2026-04-11 17:54:06

205

Views

Can this be proved using the MCT instead of the DCT?

Published on 11 Apr 2026 - 17:54

score 200 · Answer 4 · 2015-09-01 19:08:11

200

Views

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

Published on 01 Sep 2015 - 19:08

score 305 · Answer 5 · 2026-04-16 19:01:25

305

Views

Show that for every $\epsilon > 0 $ there exists $h \in \mathcal{L}^1(X)$ non-negative and $\delta > 0$ such that:

Published on 16 Apr 2026 - 19:01

score 854 · Answer 6 · 2026-04-15 03:50:41

854

Views

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but... (see below)?

Published on 15 Apr 2026 - 3:50

score 85 · Answer 7 · 2026-04-15 16:55:57

85

Views

Show $F: (0, \infty) \rightarrow \mathbb{R}$ is diff'ble

Published on 15 Apr 2026 - 16:55

score 1.3k · Answer 8 · 2026-04-17 12:42:25

1.3k

Views

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

Published on 17 Apr 2026 - 12:42

score 116 · Answer 9 · 2026-04-08 17:32:58

116

Views

Question about proof of finite additivity for measure on rectangles

Published on 08 Apr 2026 - 17:32

score 202 · Answer 10 · 2026-04-17 12:09:59

202

Views

On the integration of a Lebesgue measurable function

Published on 17 Apr 2026 - 12:09

score 7.3k · Answer 11 · 2026-04-16 08:20:33

7.3k

Views

$1/|x|^\alpha$ is integrable on the unit ball in $\mathbb R^n$ iff $\alpha < n$

Published on 16 Apr 2026 - 8:20

score 741 · Answer 12 · 2026-04-14 12:25:45

741

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Lebesgue integral on any open set is $\ge 0$, is it still $\geq 0$ on any $G_{\delta}$ set?

Published on 14 Apr 2026 - 12:25

score 1.7k · Answer 13 · 2026-04-09 03:51:43

1.7k

Views

Measure theory qualifying exam questions

Published on 09 Apr 2026 - 3:51

score 158 · Answer 14 · 2026-04-14 20:28:57

158

Views

Lebesgue Integrability of $\frac{\sin(x)}{x^a}$ on $ [0, \infty)$

Published on 14 Apr 2026 - 20:28

score 1.7k · Answer 15 · 2026-04-11 07:28:37

1.7k

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Approximate an integral with a step function

Published on 11 Apr 2026 - 7:28

List Question

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Cauchy formula for repeated Lebesgue integration

Can this be proved using the MCT instead of the DCT?

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

Show that for every $\epsilon > 0 $ there exists $h \in \mathcal{L}^1(X)$ non-negative and $\delta > 0$ such that:

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but... (see below)?

Show $F: (0, \infty) \rightarrow \mathbb{R}$ is diff'ble

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

Question about proof of finite additivity for measure on rectangles

On the integration of a Lebesgue measurable function

$1/|x|^\alpha$ is integrable on the unit ball in $\mathbb R^n$ iff $\alpha < n$

Lebesgue integral on any open set is $\ge 0$, is it still $\geq 0$ on any $G_{\delta}$ set?

Measure theory qualifying exam questions

Lebesgue Integrability of $\frac{\sin(x)}{x^a}$ on $ [0, \infty)$

Approximate an integral with a step function

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