Question

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

score 405 · Answer 1 · 2016-07-17 15:58:10

405

Views

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

Published on 17 Jul 2016 - 15:58

score 187 · Answer 2 · 2016-07-17 18:44:31

187

Views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Published on 17 Jul 2016 - 18:44

score 375 · Answer 3 · 2016-07-18 02:41:50

375

Views

$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Published on 18 Jul 2016 - 2:41

score 937 · Answer 4 · 2016-07-18 03:15:38

937

Views

Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?

Published on 18 Jul 2016 - 3:15

score 1.5k · Answer 5 · 2016-07-18 04:08:08

1.5k

Views

Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Published on 18 Jul 2016 - 4:08

score 190 · Answer 6 · 2016-07-18 18:50:57

190

Views

Does it follow that $\mu$ is a measure?

Published on 18 Jul 2016 - 18:50

score 162 · Answer 7 · 2026-04-07 03:19:14

162

Views

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Published on 07 Apr 2026 - 3:19

score 295 · Answer 8 · 2016-07-19 02:00:35

295

Views

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

Published on 19 Jul 2016 - 2:00

score 66 · Answer 9 · 2016-07-19 02:23:59

66

Views

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Published on 19 Jul 2016 - 2:23

score 609 · Answer 10 · 2026-03-29 04:41:28

609

Views

Can the Substitution Rule be Interpreted as a "Change of Measure"?

Published on 29 Mar 2026 - 4:41

score 158 · Answer 11 · 2016-07-21 14:04:19

158

Views

$\lim_{n\to\infty}\int_0^{\infty}\frac{n\sin y}{ny(1+n^2y^2)}n\,dy$ via DCT?

Published on 21 Jul 2016 - 14:04

score 474 · Answer 12 · 2016-07-21 14:56:05

474

Views

Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

Published on 21 Jul 2016 - 14:56

score 103 · Answer 13 · 2026-03-31 11:30:11

103

Views

Estimate $\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x$ without spherical coordinates.

Published on 31 Mar 2026 - 11:30

score 1.1k · Answer 14 · 2016-07-23 08:21:21

1.1k

Views

A strong version of the Dominated Convergence Theorem

Published on 23 Jul 2016 - 8:21

score 1.5k · Answer 15 · 2016-07-24 00:40:46

1.5k

Views

Sequence of integrable function with $\sum_{n=1}^\infty \|f_n\|_1<\infty$. Show that $\sum_{n=1}^\infty f_n$ converges a.e. and is integrable.

Published on 24 Jul 2016 - 0:40

List Question

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?

Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Does it follow that $\mu$ is a measure?

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Can the Substitution Rule be Interpreted as a "Change of Measure"?

$\lim_{n\to\infty}\int_0^{\infty}\frac{n\sin y}{ny(1+n^2y^2)}n\,dy$ via DCT?

Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

Estimate $\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x$ without spherical coordinates.

A strong version of the Dominated Convergence Theorem

Sequence of integrable function with $\sum_{n=1}^\infty \|f_n\|_1<\infty$. Show that $\sum_{n=1}^\infty f_n$ converges a.e. and is integrable.

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