Question

Is my understanding of Relative Topology correct for manifolds

score 33 · Answer 1 · 2026-04-11 14:44:56

33

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Is my understanding of Relative Topology correct for manifolds

Published on 11 Apr 2026 - 14:44

score 53 · Answer 2 · 2026-04-17 04:39:08

53

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Find a Borel set $F$ such that $\lim_{k \to 0} \frac{m_N(F \cap B_k(x))}{m_N(B_k(x))}$ does not exist at some $x$.

Published on 17 Apr 2026 - 4:39

score 133 · Answer 3 · 2026-04-11 06:03:52

133

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How to prove this theorem of $\sigma$-algebra?

Published on 11 Apr 2026 - 6:03

score 38 · Answer 4 · 2026-04-15 14:50:14

If $\mu(X) = \infty$ and for all $F \in {\cal E}$, there is $E \in {\cal E}$ such that $0 < \mu(E) < \mu(F)$, then $\mu$ is onto.

score 46 · Answer 5 · 2026-04-13 23:05:14

If $\nu$ and $\mu$ are finite measures on ${\cal E}$ such that $\nu(E) \le \mu(E)$ for each $E \in {\cal E}$ and $\nu(X)= \mu(X)$.

score 80 · Answer 6 · 2026-05-10 13:46:44

80

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Corollary from Hahn decomposition

Published on 10 May 2026 - 13:46

score 140 · Answer 7 · 2026-04-16 16:16:09

140

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prove that $\int_X f=\int_{[0,\infty)}\mu(\{x|f(x)>t\})dm(t)$

Published on 16 Apr 2026 - 16:16

score 1.4k · Answer 8 · 2026-04-12 19:01:45

1.4k

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Characterisation of point-mass distributions

Published on 12 Apr 2026 - 19:01

score 546 · Answer 9 · 2026-04-09 20:24:44

546

Views

Integration on manifolds?

Published on 09 Apr 2026 - 20:24

score 62 · Answer 10 · 2026-05-11 01:48:30

62

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Confusion on $\vert\vert \varphi\vert\vert_{p}$ where $p \in [1,\infty[$

Published on 11 May 2026 - 1:48

score 38 · Answer 11 · 2026-04-16 09:03:59

38

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$\mathbb{E}[h(X_n, X_{n+1})(X_{n+1} - X_n)] = 0$ for any bounded, measurable $h$ if $X_n$ is a martingale

Published on 16 Apr 2026 - 9:03

score 39 · Answer 12 · 2026-05-10 17:00:43

39

Views

How to show $\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu$

Published on 10 May 2026 - 17:00

score 266 · Answer 13 · 2026-04-12 02:45:08

266

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Property of $\mu*$-measurable sets

Published on 12 Apr 2026 - 2:45

score 570 · Answer 14 · 2026-04-16 05:29:04

570

Views

If $\nu_1 \ll \mu_1$ and $\nu_2 \ll \mu_2$ then $\nu_1 \otimes \nu_2 \ll \mu_1 \otimes \mu_2$

Published on 16 Apr 2026 - 5:29

score 59 · Answer 15 · 2026-05-14 17:12:45

59

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value function of optimal transport

Published on 14 May 2026 - 17:12

List Question

Is my understanding of Relative Topology correct for manifolds

Find a Borel set $F$ such that $\lim_{k \to 0} \frac{m_N(F \cap B_k(x))}{m_N(B_k(x))}$ does not exist at some $x$.

How to prove this theorem of $\sigma$-algebra?

If $\mu(X) = \infty$ and for all $F \in {\cal E}$, there is $E \in {\cal E}$ such that $0 < \mu(E) < \mu(F)$, then $\mu$ is onto.

If $\nu$ and $\mu$ are finite measures on ${\cal E}$ such that $\nu(E) \le \mu(E)$ for each $E \in {\cal E}$ and $\nu(X)= \mu(X)$.

Corollary from Hahn decomposition

prove that $\int_X f=\int_{[0,\infty)}\mu(\{x|f(x)>t\})dm(t)$

Characterisation of point-mass distributions

Integration on manifolds?

Confusion on $\vert\vert \varphi\vert\vert_{p}$ where $p \in [1,\infty[$

$\mathbb{E}[h(X_n, X_{n+1})(X_{n+1} - X_n)] = 0$ for any bounded, measurable $h$ if $X_n$ is a martingale

How to show $\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu$

Property of $\mu*$-measurable sets

If $\nu_1 \ll \mu_1$ and $\nu_2 \ll \mu_2$ then $\nu_1 \otimes \nu_2 \ll \mu_1 \otimes \mu_2$

value function of optimal transport

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