Question

Prove $\lim\limits_{n\to\infty}f_n'(x)=0$ for a.e. $x\in[a,b]$.

score 70 · Answer 1 · 2026-04-17 07:41:04

70

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Prove $\lim\limits_{n\to\infty}f_n'(x)=0$ for a.e. $x\in[a,b]$.

Published on 17 Apr 2026 - 7:41

score 101 · Answer 2 · 2026-04-10 20:48:24

101

Views

Norm equivalence constants

Published on 10 Apr 2026 - 20:48

score 123 · Answer 3 · 2026-04-15 08:35:12

123

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For $0 < \alpha < 1$, construct an open set $E \subset [0,1]$ which is dense in $[0,1]$ and $m(E) = \alpha$.

Published on 15 Apr 2026 - 8:35

score 58 · Answer 4 · 2026-04-15 06:42:02

58

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$L^2$ Convergence of series of normally distributed independent random variables

Published on 15 Apr 2026 - 6:42

score 29 · Answer 5 · 2026-04-14 06:04:29

29

Views

Square integrability of $ \exp \{-\int t(1-\hat{f}(s/(At)^{\beta})) \} dt$

Published on 14 Apr 2026 - 6:04

score 69 · Answer 6 · 2026-03-26 11:03:16

69

Views

Prove that if $f : M → M$ preserves a probability $\mu$, then for any $k \geq 2 $ $f^k$ preserves $\mu$

Published on 26 Mar 2026 - 11:03

score 56 · Answer 7 · 2026-04-11 00:27:09

56

Views

Set with measure zero

Published on 11 Apr 2026 - 0:27

score 182 · Answer 8 · 2026-04-14 16:23:24

182

Views

Prove that $\inf \left\{ \int_E f d\mu : \mu (E) \geq \alpha\right\} > 0$

Published on 14 Apr 2026 - 16:23

score 150 · Answer 9 · 2026-04-17 08:05:38

150

Views

Prove that if $f\in L^{p}(E), 1\leq p<\infty, m(E)<\infty$, then $f\in L^{q}(E)$ for all $1\leq q\leq p$

Published on 17 Apr 2026 - 8:05

score 68 · Answer 10 · 2026-04-15 03:58:12

68

Views

If f > 0, is ∫ f > 0?

Published on 15 Apr 2026 - 3:58

score 170 · Answer 11 · 2026-04-16 08:50:25

170

Views

Proving that $L^{p}(\mathbb{R}^d)\not\subset L^{\infty}(\mathbb{R}^d)$ for $1\leq p<\infty.$

Published on 16 Apr 2026 - 8:50

score 73 · Answer 12 · 2026-04-16 08:31:45

73

Views

Prove $\int \bigg(\sum_{n=1}^{\infty} f_n \bigg) d \mu = \sum_{n=1}^{\infty} \bigg( \int f_n d \mu \bigg) $

Published on 16 Apr 2026 - 8:31

score 400 · Answer 13 · 2026-03-26 08:55:19

400

Views

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Published on 26 Mar 2026 - 8:55

score 112 · Answer 14 · 2026-05-16 22:24:52

112

Views

Pushforward Measure Continuously Differentiable Function

Published on 16 May 2026 - 22:24

score 31 · Answer 15 · 2026-04-17 11:29:31

31

Views

If $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ prove $d_f (t) \leq \bigg( \frac{||{f}||_{L^p(X)}}{t} \bigg)^p.$

Published on 17 Apr 2026 - 11:29

List Question

Prove $\lim\limits_{n\to\infty}f_n'(x)=0$ for a.e. $x\in[a,b]$.

Norm equivalence constants

For $0 < \alpha < 1$, construct an open set $E \subset [0,1]$ which is dense in $[0,1]$ and $m(E) = \alpha$.

$L^2$ Convergence of series of normally distributed independent random variables

Square integrability of $ \exp \{-\int t(1-\hat{f}(s/(At)^{\beta})) \} dt$

Prove that if $f : M → M$ preserves a probability $\mu$, then for any $k \geq 2 $ $f^k$ preserves $\mu$

Set with measure zero

Prove that $\inf \left\{ \int_E f d\mu : \mu (E) \geq \alpha\right\} > 0$

Prove that if $f\in L^{p}(E), 1\leq p<\infty, m(E)<\infty$, then $f\in L^{q}(E)$ for all $1\leq q\leq p$

If f > 0, is ∫ f > 0?

Proving that $L^{p}(\mathbb{R}^d)\not\subset L^{\infty}(\mathbb{R}^d)$ for $1\leq p<\infty.$

Prove $\int \bigg(\sum_{n=1}^{\infty} f_n \bigg) d \mu = \sum_{n=1}^{\infty} \bigg( \int f_n d \mu \bigg) $

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Pushforward Measure Continuously Differentiable Function

If $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ prove $d_f (t) \leq \bigg( \frac{||{f}||_{L^p(X)}}{t} \bigg)^p.$

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