Question

Prove convergence in $L^1$

score 107 · Answer 1 · 2026-04-15 11:31:51

107

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Prove convergence in $L^1$

Published on 15 Apr 2026 - 11:31

score 107 · Answer 2 · 2026-04-10 18:02:41

107

Views

For every open subset of $\mathbb{R}^n$ there is a countable non-overlapping union of closed squares

Published on 10 Apr 2026 - 18:02

score 2.4k · Answer 3 · 2026-04-02 14:42:26

2.4k

Views

Is it always true that the Lebesgue integral of a continuous function is equal to the Riemann integral (even if they are both unbounded)?

Published on 02 Apr 2026 - 14:42

score 102 · Answer 4 · 2026-04-22 21:28:00

102

Views

Prove $\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$

Published on 22 Apr 2026 - 21:28

score 297 · Answer 5 · 2026-04-22 11:13:34

297

Views

$L^p$ space and continuous injection

Published on 22 Apr 2026 - 11:13

score 113 · Answer 6 · 2026-04-11 19:12:14

113

Views

Prove sum is commutative with measure theory result

Published on 11 Apr 2026 - 19:12

score 468 · Answer 7 · 2026-04-15 11:46:13

468

Views

Lebesgue monotone convergence theorem

Published on 15 Apr 2026 - 11:46

score 280 · Answer 8 · 2026-04-22 11:37:51

280

Views

A bounded family of functions in $L^p[E]$, where E is a measurable set, is uniformly integrable.

Published on 22 Apr 2026 - 11:37

score 261 · Answer 9 · 2026-04-12 02:45:33

261

Views

Prove a function to be analytic by dominated convergence theorem

Published on 12 Apr 2026 - 2:45

score 1.3k · Answer 10 · 2026-04-16 13:45:53

1.3k

Views

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable.

Published on 16 Apr 2026 - 13:45

score 891 · Answer 11 · 2026-04-15 20:20:24

891

Views

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$

Published on 15 Apr 2026 - 20:20

score 1.2k · Answer 12 · 2026-04-14 04:55:42

1.2k

Views

Let $f$ be integrable over $\mathbb{R}$. Show that the following four assertions are equivalent:

Published on 14 Apr 2026 - 4:55

score 2k · Answer 13 · 2026-04-13 06:48:34

2k

Views

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions.

Published on 13 Apr 2026 - 6:48

score 56 · Answer 14 · 2026-04-15 01:11:24

56

Views

Show that $f_n\to f$ in the norm $L^1(\mathbb{R})$ for $f\in L^1(\mathbb{R})$.

Published on 15 Apr 2026 - 1:11

score 234 · Answer 15 · 2026-03-25 23:13:18

234

Views

Eigenvalues of an integral operator

Published on 25 Mar 2026 - 23:13

List Question

Prove convergence in $L^1$

For every open subset of $\mathbb{R}^n$ there is a countable non-overlapping union of closed squares

Is it always true that the Lebesgue integral of a continuous function is equal to the Riemann integral (even if they are both unbounded)?

Prove $\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$

$L^p$ space and continuous injection

Prove sum is commutative with measure theory result

Lebesgue monotone convergence theorem

A bounded family of functions in $L^p[E]$, where E is a measurable set, is uniformly integrable.

Prove a function to be analytic by dominated convergence theorem

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable.

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$

Let $f$ be integrable over $\mathbb{R}$. Show that the following four assertions are equivalent:

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions.

Show that $f_n\to f$ in the norm $L^1(\mathbb{R})$ for $f\in L^1(\mathbb{R})$.

Eigenvalues of an integral operator

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