Question

Let $f_n=\sqrt{n}e^{-nx}$ on $(0,1)$ does $\int_{0}^{1}f_nh \to 0$ for all $h \in L^1$

score 37 · Answer 1 · 2020-04-22 20:48:31

37

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Let $f_n=\sqrt{n}e^{-nx}$ on $(0,1)$ does $\int_{0}^{1}f_nh \to 0$ for all $h \in L^1$

Published on 22 Apr 2020 - 20:48

score 75 · Answer 2 · 2026-03-25 16:13:47

75

Views

Reciprocal of Hölder's inequality

Published on 25 Mar 2026 - 16:13

score 130 · Answer 3 · 2020-04-23 00:13:55

130

Views

$\varphi$ monotone,continuous on $[0,1]$. Show $\varphi$ is AC, iff for any borel $f_n\to f$ in $L^1$,$\{\varphi(f_n)\}$ converges in measure.

Published on 23 Apr 2020 - 0:13

score 57 · Answer 4 · 2020-04-23 10:50:25

57

Views

Why doesn't $f_n(x)=(n^2x^2)/\left(1+x^2\right)^n$ verify $\lim_{n\to\infty}\int f_n = \int\lim_{n\to\infty} f_n$?

Published on 23 Apr 2020 - 10:50

score 374 · Answer 5 · 2020-04-23 12:00:54

374

Views

A Question Lebesgue Dominated Convergence Theorem

Published on 23 Apr 2020 - 12:00

score 66 · Answer 6 · 2026-03-29 03:03:56

66

Views

Can we define $N_t:=\int_0^t \operatorname{sign}(B_s) \mathrm{d}s$?

Published on 29 Mar 2026 - 3:03

score 125 · Answer 7 · 2026-03-25 15:40:05

125

Views

Understanding Almost Everywhere Statements in Stromberg's Book

Published on 25 Mar 2026 - 15:40

score 180 · Answer 8 · 2020-04-23 23:50:21

180

Views

Different and Alternative Proofs of Dominated Convergence Theorem

Published on 23 Apr 2020 - 23:50

score 48 · Answer 9 · 2020-04-24 00:27:47

48

Views

Let $m$ be Lebesgue measure on $[0,1]$ and define $\mu(E)=m(E)+im(E\cap[0,\frac12]).$

Published on 24 Apr 2020 - 0:27

score 438 · Answer 10 · 2026-02-24 05:32:50

438

Views

Riemann integral counterexample to dominated convergence theorem?

Published on 24 Feb 2026 - 5:32

score 352 · Answer 11 · 2026-02-23 15:09:43

352

Views

Radon Nikodym derivative of a restriction of a measure.

Published on 23 Feb 2026 - 15:09

score 31 · Answer 12 · 2020-04-24 23:35:09

31

Views

Integrating $\chi_\mathbb{Q}e^{\|x|}$ over counting measure.

Published on 24 Apr 2020 - 23:35

score 851 · Answer 13 · 2020-04-24 23:55:11

851

Views

Example of a bounded linear functional $T:L^ \infty \to \mathbb{R}$ which cannot be expressed by an integral $\int_{[a,b]}fg$ for an integrable $f$

Published on 24 Apr 2020 - 23:55

score 374 · Answer 14 · 2026-03-22 11:16:07

374

Views

Push-Forward with Multiple Random Variables

Published on 22 Mar 2026 - 11:16

score 129 · Answer 15 · 2020-04-25 02:18:44

129

Views

Lebesgue points and convolution

Published on 25 Apr 2020 - 2:18

List Question

Let $f_n=\sqrt{n}e^{-nx}$ on $(0,1)$ does $\int_{0}^{1}f_nh \to 0$ for all $h \in L^1$

Reciprocal of Hölder's inequality

$\varphi$ monotone,continuous on $[0,1]$. Show $\varphi$ is AC, iff for any borel $f_n\to f$ in $L^1$,$\{\varphi(f_n)\}$ converges in measure.

Why doesn't $f_n(x)=(n^2x^2)/\left(1+x^2\right)^n$ verify $\lim_{n\to\infty}\int f_n = \int\lim_{n\to\infty} f_n$?

A Question Lebesgue Dominated Convergence Theorem

Can we define $N_t:=\int_0^t \operatorname{sign}(B_s) \mathrm{d}s$?

Understanding Almost Everywhere Statements in Stromberg's Book

Different and Alternative Proofs of Dominated Convergence Theorem

Let $m$ be Lebesgue measure on $[0,1]$ and define $\mu(E)=m(E)+im(E\cap[0,\frac12]).$

Riemann integral counterexample to dominated convergence theorem?

Radon Nikodym derivative of a restriction of a measure.

Integrating $\chi_\mathbb{Q}e^{\|x|}$ over counting measure.

Example of a bounded linear functional $T:L^ \infty \to \mathbb{R}$ which cannot be expressed by an integral $\int_{[a,b]}fg$ for an integrable $f$

Push-Forward with Multiple Random Variables

Lebesgue points and convolution

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