List questions from lebesgue-integral

Published on 01 Apr 2026 - 11:02

46

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Reflection Property of the Lebesgue Integral

Published on 09 May 2023 - 22:01

106

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Is it true that $\min\{h, f_n\}\to h$ in measure provided $f_n\to f$ in measure?

Published on 11 Apr 2026 - 7:26

46

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Show that $\int_E |f|^q \leq CN^{q/p} |E|^{1-q/p}$.

Published on 10 May 2023 - 10:32

61

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Lebesgue integral preserves the inequality

Published on 11 Apr 2026 - 20:45

60

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Approximating the exponential function with simple functions

Published on 25 Mar 2026 - 9:22

29

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Let $E\subset \Bbb R^n$ be measurable, $mE>0$. and $0<p_0\neq q_0<\infty$, $L^{p_0}(E)\subset L^{q_0}(E)$. Show $q_0<p_0$.

Published on 12 May 2023 - 12:49

54

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Change of variables in Lebesgue Integral, 2nd question

Published on 09 Apr 2026 - 22:55

26

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whether we can put the limit inside the $L^\infty$ norm

Published on 09 Apr 2026 - 11:14

333

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Is $x^a $sin$(x^{-a})$ Hölder continuous?

Published on 10 Apr 2026 - 8:03

17

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Show that $E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt)$

Published on 12 Apr 2026 - 7:33

28

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Prove that $E(F(X))=\frac{1}{2}+\frac{1}{2}\sum_x[P(X=x)]^2$

Published on 12 Apr 2026 - 7:34

49

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Double integral of $f(x,y)=\frac{xy}{(1-|x|)^2+(1-|y|)^2)}$ on $[-1,1] \times [-1,1]$

Published on 14 May 2023 - 14:28

52

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If $f\in L^{1}([0,1]\times[0,1])$ and $F(x)=\int_{[0,x]\times[0,x]} f$ for$x\in [0,1]$, prove $F$ is absolutely continuous from the definition.

Published on 11 Apr 2026 - 17:19

46

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Prove that $\int_{X>\alpha}XdP\le\alpha(-\log F(\alpha))+\int_\alpha^\infty(-\log F(t))\lambda(dt)$

Published on 12 Apr 2026 - 7:41

List Question

Variance of A sin(Fx)+Bx, and correlation

Reflection Property of the Lebesgue Integral

Is it true that $\min\{h, f_n\}\to h$ in measure provided $f_n\to f$ in measure?

Show that $\int_E |f|^q \leq CN^{q/p} |E|^{1-q/p}$.

Lebesgue integral preserves the inequality

Approximating the exponential function with simple functions

Let $E\subset \Bbb R^n$ be measurable, $mE>0$. and $0<p_0\neq q_0<\infty$, $L^{p_0}(E)\subset L^{q_0}(E)$. Show $q_0<p_0$.

Change of variables in Lebesgue Integral, 2nd question

whether we can put the limit inside the $L^\infty$ norm

Is $x^a $sin$(x^{-a})$ Hölder continuous?

Show that $E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt)$

Prove that $E(F(X))=\frac{1}{2}+\frac{1}{2}\sum_x[P(X=x)]^2$

Double integral of $f(x,y)=\frac{xy}{(1-|x|)^2+(1-|y|)^2)}$ on $[-1,1] \times [-1,1]$

If $f\in L^{1}([0,1]\times[0,1])$ and $F(x)=\int_{[0,x]\times[0,x]} f$ for$x\in [0,1]$, prove $F$ is absolutely continuous from the definition.

Prove that $\int_{X>\alpha}XdP\le\alpha(-\log F(\alpha))+\int_\alpha^\infty(-\log F(t))\lambda(dt)$

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