List questions from lebesgue-integral

Published on 13 Apr 2026 - 15:01

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Evaluate the limit of $\lim_{n \to \infty}n.\mu(X_n)$

Published on 16 Apr 2026 - 14:13

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If $f(x,y)=e^{-xy}\sin x\sin y$, $\int_0^\infty\int_0^\infty f(x,y)dxdy$ exists but $\iint_{\mathbb{R}_{\geq 0}^2} f(x,y) dxdy$ does not.

Published on 18 Apr 2026 - 23:16

6.7k

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Does absolute integrability imply integrability?

Published on 12 Apr 2026 - 5:12

193

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Proving Fourier transform is continuous with limited knowledge.

Published on 14 Apr 2026 - 4:47

1.9k

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Adjoint of an operator on $L^2$

Published on 13 Apr 2026 - 4:01

71

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Let $\{a_n\}_{n=1}^{\infty}\subset [0,1]$ be equidistributed and set $\mu_N=\tfrac 1N\sum_{n=1}^N\delta_{a_n}$. Then $\int_0^1fd\mu_N\to\int_0^1fdm$

Published on 11 May 2026 - 8:42

96

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Lebesgue Measure Theory - Constructing a sequence of non-negative simple functions $\phi_{n}$ whose limit is $f$.

Published on 16 Apr 2026 - 2:55

108

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Fatou's lemma. Prove or disprove.

Published on 16 Apr 2026 - 3:25

63

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Show that $\|\check{f_k}\|_1 \to \infty$ as $k \to \infty$

Published on 13 Apr 2026 - 12:33

39

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For $L^q$ functions does convergence to zero in $L^p$ imply convergence to zero in $L^q$, when $p<q$?

Published on 16 Apr 2026 - 5:48

843

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Proving Van der Waerden’s Example of a Continuous Nowhere Differentiable Function

Published on 15 Apr 2026 - 10:34

61

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Proving $L^2$ boundedness

Published on 16 Apr 2026 - 4:51

97

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Convergence in $L^2(1,\infty)$ – counterexample

Published on 15 Apr 2026 - 17:26

507

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Substitution rule in the sense of Lebesgue integral

Published on 11 Apr 2026 - 17:12

List Question

Lebesgue sets of functions

Evaluate the limit of $\lim_{n \to \infty}n.\mu(X_n)$

If $f(x,y)=e^{-xy}\sin x\sin y$, $\int_0^\infty\int_0^\infty f(x,y)dxdy$ exists but $\iint_{\mathbb{R}_{\geq 0}^2} f(x,y) dxdy$ does not.

Does absolute integrability imply integrability?

Proving Fourier transform is continuous with limited knowledge.

Adjoint of an operator on $L^2$

Let $\{a_n\}_{n=1}^{\infty}\subset [0,1]$ be equidistributed and set $\mu_N=\tfrac 1N\sum_{n=1}^N\delta_{a_n}$. Then $\int_0^1fd\mu_N\to\int_0^1fdm$

Lebesgue Measure Theory - Constructing a sequence of non-negative simple functions $\phi_{n}$ whose limit is $f$.

Fatou's lemma. Prove or disprove.

Show that $\|\check{f_k}\|_1 \to \infty$ as $k \to \infty$

For $L^q$ functions does convergence to zero in $L^p$ imply convergence to zero in $L^q$, when $p<q$?

Proving Van der Waerden’s Example of a Continuous Nowhere Differentiable Function

Proving $L^2$ boundedness

Convergence in $L^2(1,\infty)$ – counterexample

Substitution rule in the sense of Lebesgue integral

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