Question

Motivation and references for an integral formula in measure theory

score 362 · Answer 1 · 2019-03-25 11:43:03

362

Views

Motivation and references for an integral formula in measure theory

Published on 25 Mar 2019 - 11:43

score 107 · Answer 2 · 2019-03-27 06:59:22

107

Views

Why this highly oscillatory function is not Lebesgue integrable?

Published on 27 Mar 2019 - 6:59

score 339 · Answer 3 · 2019-03-28 13:58:49

339

Views

Show that $\lim_{n\to \infty} \int_E \cos^2(nx + a_n) dx = \frac{1}{2}m(E)$

Published on 28 Mar 2019 - 13:58

score 70 · Answer 4 · 2019-03-28 17:23:09

70

Views

Consider $f_n, f \in L^2(d\mu)$, $f_n(x) \to f(x)$ a.e. and $\|f_n\|_2 \to \|f\|_2$. Use Egorov's theorem to show that $f_n \to f$ in $L^2(d\mu)$.

Published on 28 Mar 2019 - 17:23

score 2k · Answer 5 · 2019-03-31 00:50:15

2k

Views

Hardy-Littlewood Maximal Function and Characteristic Functions

Published on 31 Mar 2019 - 0:50

score 70 · Answer 6 · 2026-03-25 23:42:19

70

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A short way to determine the divergence of the integral: $\int\limits_0^1 \left| \frac{1}{x}\cos\left(\frac{1}{x}\right) \right| dx$

Published on 25 Mar 2026 - 23:42

score 1k · Answer 7 · 2019-03-31 01:14:15

1k

Views

A small doubt about the dominated convergence theorem

Published on 31 Mar 2019 - 1:14

score 453 · Answer 8 · 2019-04-02 02:42:01

453

Views

Showing that $\lim_{n \to \infty} \int_E \cos(nx) = \lim_{n \to \infty} \int_E \sin(nx) = 0$

Published on 02 Apr 2019 - 2:42

score 121 · Answer 9 · 2026-03-25 15:45:41

121

Views

Show that $\phi (\lambda^{d})=\lambda^{d}$

Published on 25 Mar 2026 - 15:45

score 216 · Answer 10 · 2019-04-02 14:42:10

216

Views

Direct sum of Sobolev spaces

Published on 02 Apr 2019 - 14:42

score 126 · Answer 11 · 2019-04-02 16:08:50

126

Views

$f_n(x) \to f(x)$ and $\int |f_n|^2 \, d\mu \to \int |f|^2 \, d\mu$. Use Egorov's theorem to show $f_n \to f$ in $L^2(d\mu)$.

Published on 02 Apr 2019 - 16:08

score 78 · Answer 12 · 2026-03-25 14:35:50

78

Views

If $f$ is nondecreasing and $h$ is of bounded variation with $|h(t)-h(s)|≤C\sqrt{f(t)-f(s)}$, then $\int|X||{\rm d}h|≤C\sqrt{\int|X|^2\:{\rm d}f}$

Published on 25 Mar 2026 - 14:35

score 49 · Answer 13 · 2019-04-02 19:49:24

49

Views

Let $g: \mathbb{R} \to \overline{\mathbb{R}}$ be integrable. If $\int_K g \, d m = 0$ for every compact $K \subset \mathbb{R}$, then $g = 0$ a.e.

Published on 02 Apr 2019 - 19:49

score 385 · Answer 14 · 2019-04-03 19:41:31

385

Views

Example that does not contradict the Dominated Convergence Theorem

Published on 03 Apr 2019 - 19:41

score 63 · Answer 15 · 2019-04-03 23:14:03

63

Views

Does an example of a function exist where the Lebesgue Integral doesn't work?

Published on 03 Apr 2019 - 23:14

List Question

Motivation and references for an integral formula in measure theory

Why this highly oscillatory function is not Lebesgue integrable?

Show that $\lim_{n\to \infty} \int_E \cos^2(nx + a_n) dx = \frac{1}{2}m(E)$

Consider $f_n, f \in L^2(d\mu)$, $f_n(x) \to f(x)$ a.e. and $\|f_n\|_2 \to \|f\|_2$. Use Egorov's theorem to show that $f_n \to f$ in $L^2(d\mu)$.

Hardy-Littlewood Maximal Function and Characteristic Functions

A short way to determine the divergence of the integral: $\int\limits_0^1 \left| \frac{1}{x}\cos\left(\frac{1}{x}\right) \right| dx$

A small doubt about the dominated convergence theorem

Showing that $\lim_{n \to \infty} \int_E \cos(nx) = \lim_{n \to \infty} \int_E \sin(nx) = 0$

Show that $\phi (\lambda^{d})=\lambda^{d}$

Direct sum of Sobolev spaces

$f_n(x) \to f(x)$ and $\int |f_n|^2 \, d\mu \to \int |f|^2 \, d\mu$. Use Egorov's theorem to show $f_n \to f$ in $L^2(d\mu)$.

If $f$ is nondecreasing and $h$ is of bounded variation with $|h(t)-h(s)|≤C\sqrt{f(t)-f(s)}$, then $\int|X||{\rm d}h|≤C\sqrt{\int|X|^2\:{\rm d}f}$

Let $g: \mathbb{R} \to \overline{\mathbb{R}}$ be integrable. If $\int_K g \, d m = 0$ for every compact $K \subset \mathbb{R}$, then $g = 0$ a.e.

Example that does not contradict the Dominated Convergence Theorem

Does an example of a function exist where the Lebesgue Integral doesn't work?

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