Question

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

score 131 · Answer 1 · 2026-04-23 22:20:18

131

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Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

Published on 23 Apr 2026 - 22:20

score 699 · Answer 2 · 2026-04-16 15:38:19

699

Views

Show that there exists a continuous function $f$ such that $\int |\chi_A-f| d\lambda\lt \epsilon$

Published on 16 Apr 2026 - 15:38

score 1k · Answer 3 · 2026-04-09 16:08:36

1k

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Markov Inequality proof (measure theory)

Published on 09 Apr 2026 - 16:08

score 243 · Answer 4 · 2026-04-10 02:42:58

243

Views

Function of bounded variation and integration

Published on 10 Apr 2026 - 2:42

score 604 · Answer 5 · 2026-04-16 23:06:37

604

Views

Application of Fubini-Tonelli's Theorem on function $\frac{2}{\pi}e^{-ax}\cos(x\cos{\theta})$

Published on 16 Apr 2026 - 23:06

score 164 · Answer 6 · 2026-04-16 20:56:53

164

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$f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$?

Published on 16 Apr 2026 - 20:56

score 340 · Answer 7 · 2026-04-16 08:48:05

340

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Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$.

Published on 16 Apr 2026 - 8:48

score 69 · Answer 8 · 2026-04-14 22:22:31

69

Views

Show $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$

Published on 14 Apr 2026 - 22:22

score 59 · Answer 9 · 2026-04-04 17:08:07

59

Views

Lebesgue measurable integration, density

Published on 04 Apr 2026 - 17:08

score 52 · Answer 10 · 2026-04-11 13:47:06

52

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Monotonic increasing and convergence in measure

Published on 11 Apr 2026 - 13:47

score 6.4k · Answer 11 · 2026-04-25 07:32:17

6.4k

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How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

Published on 25 Apr 2026 - 7:32

score 191 · Answer 12 · 2026-03-28 22:26:32

191

Views

When do we have the formula $f(t)=e^{\lambda t}f(0)+\int_0^te^{\lambda (t-s)}g(s)ds$?

Published on 28 Mar 2026 - 22:26

score 92 · Answer 13 · 2026-04-16 13:47:25

92

Views

Evaluate $\int_{0}^{\infty} \frac{\sinh bx}{\sinh ax} dx $

Published on 16 Apr 2026 - 13:47

score 197 · Answer 14 · 2026-04-16 00:33:29

197

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Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

Published on 16 Apr 2026 - 0:33

score 50 · Answer 15 · 2026-04-16 01:03:43

Show $\lim\limits_k \int_{A_k} f_k = \lim\limits_k\int_A f_k,\;$ given $f_k \in\mathcal{L}^1(\mathbb{R}^n),\; \lim\limits_k\lambda(A_k\Delta A) = 0$.

List Question

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

Show that there exists a continuous function $f$ such that $\int |\chi_A-f| d\lambda\lt \epsilon$

Markov Inequality proof (measure theory)

Function of bounded variation and integration

Application of Fubini-Tonelli's Theorem on function $\frac{2}{\pi}e^{-ax}\cos(x\cos{\theta})$

$f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$?

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$.

Show $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$

Lebesgue measurable integration, density

Monotonic increasing and convergence in measure

How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

When do we have the formula $f(t)=e^{\lambda t}f(0)+\int_0^te^{\lambda (t-s)}g(s)ds$?

Evaluate $\int_{0}^{\infty} \frac{\sinh bx}{\sinh ax} dx $

Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

Show $\lim\limits_k \int_{A_k} f_k = \lim\limits_k\int_A f_k,\;$ given $f_k \in\mathcal{L}^1(\mathbb{R}^n),\; \lim\limits_k\lambda(A_k\Delta A) = 0$.

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