Question

A probability measure is nonatomic iff its distribution function is continuous

score 2.3k · Answer 1 · 2026-04-17 01:06:07

2.3k

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A probability measure is nonatomic iff its distribution function is continuous

Published on 17 Apr 2026 - 1:06

score 107 · Answer 2 · 2026-04-12 16:50:32

107

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Integral formula containing cumulative distribution function

Published on 12 Apr 2026 - 16:50

score 1k · Answer 3 · 2026-04-17 03:45:50

1k

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Prove that $\int f\,d\mu>0 \text{ if and only if } \mu(\{x\in X: f(x)>0\})>0.$

Published on 17 Apr 2026 - 3:45

score 73 · Answer 4 · 2026-04-13 17:12:19

73

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Probablity of a tie for iid continuous r.v.s

Published on 13 Apr 2026 - 17:12

score 276 · Answer 5 · 2026-04-16 03:41:54

Prove that $\int \vert \ \vert f_n\vert-\vert f_n-f\vert^p -\vert f\ \vert \ \vert d\mu \to 0$, if $\sup_n\vert f_n \vert\leq C$

score 193 · Answer 6 · 2026-04-16 19:08:47

Show that for sequence of independent random variables $X_1,X_2,...$ $\sum_{i=1}^\infty X_i$ either converges or diverges almost surely

score 230 · Answer 7 · 2026-04-13 05:00:26

230

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Application of the the Dominated Convergence Theorem

Published on 13 Apr 2026 - 5:00

score 201 · Answer 8 · 2026-04-09 17:15:26

201

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If $\int f_nd\mu \to \int f d \mu <\infty$ then for all measurable $E$, $\int _E f_nd\mu \to \int _E f d\mu$

Published on 09 Apr 2026 - 17:15

score 238 · Answer 9 · 2026-04-08 17:17:21

238

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Measure theory and integration textbook

Published on 08 Apr 2026 - 17:17

score 181 · Answer 10 · 2026-04-16 15:07:28

181

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Prove that $\mathscr{B}(\mathbb{R}^d)$ is a subset of $\mathscr{F}$, a $\sigma$-field

Published on 16 Apr 2026 - 15:07

score 105 · Answer 11 · 2026-04-14 01:26:40

105

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Uniqueness for classical solution of PDE

Published on 14 Apr 2026 - 1:26

score 161 · Answer 12 · 2026-04-17 07:01:36

Let $X$ be positive r.v. Prove $\lim_{x\to\infty} xE[X_{\left\{ X > x \right\}}^{-1}] = \lim_{x\downarrow 0} xE[X^{-1}1_{\left\{ X > x \right\}}]=0.$

score 193 · Answer 13 · 2026-04-15 10:43:55

193

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Equivalent definition of a universal measure zero set.

Published on 15 Apr 2026 - 10:43

score 402 · Answer 14 · 2026-04-13 07:05:00

402

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$A+B$ Borel set if $A$ is countable, open.

Published on 13 Apr 2026 - 7:05

score 114 · Answer 15 · 2026-04-11 10:55:54

114

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Prove $f(x,y)$ defined on $\mathbb{R}^2$ is Lebesgue Measurable if $f$ is continuous in each variable separately.

Published on 11 Apr 2026 - 10:55

List Question

A probability measure is nonatomic iff its distribution function is continuous

Integral formula containing cumulative distribution function

Prove that $\int f\,d\mu>0 \text{ if and only if } \mu(\{x\in X: f(x)>0\})>0.$

Probablity of a tie for iid continuous r.v.s

Prove that $\int \vert \ \vert f_n\vert-\vert f_n-f\vert^p -\vert f\ \vert \ \vert d\mu \to 0$, if $\sup_n\vert f_n \vert\leq C$

Show that for sequence of independent random variables $X_1,X_2,...$ $\sum_{i=1}^\infty X_i$ either converges or diverges almost surely

Application of the the Dominated Convergence Theorem

If $\int f_nd\mu \to \int f d \mu <\infty$ then for all measurable $E$, $\int _E f_nd\mu \to \int _E f d\mu$

Measure theory and integration textbook

Prove that $\mathscr{B}(\mathbb{R}^d)$ is a subset of $\mathscr{F}$, a $\sigma$-field

Uniqueness for classical solution of PDE

Let $X$ be positive r.v. Prove $\lim_{x\to\infty} xE[X_{\left\{ X > x \right\}}^{-1}] = \lim_{x\downarrow 0} xE[X^{-1}1_{\left\{ X > x \right\}}]=0.$

Equivalent definition of a universal measure zero set.

$A+B$ Borel set if $A$ is countable, open.

Prove $f(x,y)$ defined on $\mathbb{R}^2$ is Lebesgue Measurable if $f$ is continuous in each variable separately.

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