Question

Approximating Lebesgue integrable functions with continuous $2\pi$ periodic functions and trig polynomials

score 144 · Answer 1 · 2018-04-25 00:01:37

144

Views

Approximating Lebesgue integrable functions with continuous $2\pi$ periodic functions and trig polynomials

Published on 25 Apr 2018 - 0:01

score 276 · Answer 2 · 2018-04-25 00:13:52

276

Views

Measurable function is integrable if and only if this serie is convergent

Published on 25 Apr 2018 - 0:13

score 121 · Answer 3 · 2018-04-25 13:29:27

121

Views

Existence of Lebesgue integral with counting measure

Published on 25 Apr 2018 - 13:29

score 174 · Answer 4 · 2018-04-25 17:35:13

174

Views

Show $ \lim_{n \rightarrow \infty} \int_{\Omega} f_n\,d\mu_n = \int_{\Omega} f\,d\mu $

Published on 25 Apr 2018 - 17:35

score 97 · Answer 5 · 2018-04-26 21:26:14

97

Views

Prove these can't both be true: $\int_{0}^{\pi} [f(x) - \sin x] ^2 \, dx \le \frac{4}{9}$, $\int_{0}^{\pi} [f(x) - \cos x] ^2 \, dx \le \frac{1}{9}$

Published on 26 Apr 2018 - 21:26

score 30 · Answer 6 · 2018-04-26 23:42:06

30

Views

Two questions regarding Lebesgue measure and integral

Published on 26 Apr 2018 - 23:42

score 34 · Answer 7 · 2018-04-27 19:16:02

34

Views

application for Integration of non-negative measurable functions

Published on 27 Apr 2018 - 19:16

score 2.5k · Answer 8 · 2018-04-28 14:51:14

2.5k

Views

Is Lebesgue integral w.r.t. counting measure the same thing as sum (on an arbitrary set)?

Published on 28 Apr 2018 - 14:51

score 43 · Answer 9 · 2018-04-28 16:12:09

43

Views

Help bounding this function to apply dominated convergence theorem

Published on 28 Apr 2018 - 16:12

score 253 · Answer 10 · 2026-03-25 13:55:22

253

Views

Show that $g\in L^q(E)$ and $\|g\|_q\leq M.$

Published on 25 Mar 2026 - 13:55

score 45 · Answer 11 · 2018-04-29 21:43:21

45

Views

$\int_{\mathbb{T}} f(x) dx = \int_{\mathbb{T}} f(x-x_0) dx$ for $f \in C_0^{\infty}( \omega), \omega\subset \mathbb{T}$?

Published on 29 Apr 2018 - 21:43

score 346 · Answer 12 · 2018-04-30 13:40:14

346

Views

Prove that $f^2$ is Lebesgue integrable if and only if $\sum_{k=1}^\infty k \cdot m\{x\in A: |f(x)|>k\}<\infty.$

Published on 30 Apr 2018 - 13:40

score 53 · Answer 13 · 2018-04-30 18:19:25

53

Views

Show that the following set is dense in $L^1 (\Bbb R^d)$.

Published on 30 Apr 2018 - 18:19

score 354 · Answer 14 · 2018-05-01 19:58:30

354

Views

Show that $f'$ is square integrable.

Published on 01 May 2018 - 19:58

score 98 · Answer 15 · 2018-05-02 17:12:08

98

Views

Prove or disprove: if $\mu(E) > 0$, then $\int_E \int_E \frac{dx\,dy}{|x-y|} = \infty.$

Published on 02 May 2018 - 17:12

List Question

Approximating Lebesgue integrable functions with continuous $2\pi$ periodic functions and trig polynomials

Measurable function is integrable if and only if this serie is convergent

Existence of Lebesgue integral with counting measure

Show $ \lim_{n \rightarrow \infty} \int_{\Omega} f_n\,d\mu_n = \int_{\Omega} f\,d\mu $

Prove these can't both be true: $\int_{0}^{\pi} [f(x) - \sin x] ^2 \, dx \le \frac{4}{9}$, $\int_{0}^{\pi} [f(x) - \cos x] ^2 \, dx \le \frac{1}{9}$

Two questions regarding Lebesgue measure and integral

application for Integration of non-negative measurable functions

Is Lebesgue integral w.r.t. counting measure the same thing as sum (on an arbitrary set)?

Help bounding this function to apply dominated convergence theorem

Show that $g\in L^q(E)$ and $\|g\|_q\leq M.$

$\int_{\mathbb{T}} f(x) dx = \int_{\mathbb{T}} f(x-x_0) dx$ for $f \in C_0^{\infty}( \omega), \omega\subset \mathbb{T}$?

Prove that $f^2$ is Lebesgue integrable if and only if $\sum_{k=1}^\infty k \cdot m\{x\in A: |f(x)|>k\}<\infty.$

Show that the following set is dense in $L^1 (\Bbb R^d)$.

Show that $f'$ is square integrable.

Prove or disprove: if $\mu(E) > 0$, then $\int_E \int_E \frac{dx\,dy}{|x-y|} = \infty.$

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