Question

Show that $f_n\overset{L_p}{\longrightarrow}f\Rightarrow f_n\overset{\mu}{\longrightarrow}f$

score 29 · Answer 1 · 2023-05-11 08:45:16

29

Views

Show that $f_n\overset{L_p}{\longrightarrow}f\Rightarrow f_n\overset{\mu}{\longrightarrow}f$

Published on 11 May 2023 - 8:45

score 34 · Answer 2 · 2023-05-14 13:39:01

34

Views

Show that $P_n\Rightarrow P$ if and only if $P_n(x_k)\rightarrow P(x_k)$ for every $k\ge$ and $\sum_{k=1}^\infty P(x_k)=1$

Published on 14 May 2023 - 13:39

score 101 · Answer 3 · 2026-03-25 11:08:50

101

Views

Elementary functions are measurable

Published on 25 Mar 2026 - 11:08

score 50 · Answer 4 · 2026-03-25 11:11:51

50

Views

Right-continuous, real valued functions of a real variable are $\mathcal{B}$-measurable

Published on 25 Mar 2026 - 11:11

score 44 · Answer 5 · 2023-05-28 15:20:40

44

Views

Prove that $\sup\limits_nf_n$ and $\inf\limits_nf_n$ are measurable.

Published on 28 May 2023 - 15:20

score 46 · Answer 6 · 2026-03-27 13:45:43

46

Views

Proving lower bound for an integral

Published on 27 Mar 2026 - 13:45

score 53 · Answer 7 · 2023-06-07 01:33:52

53

Views

How to understand that a function is measurable for a.e. x?

Published on 07 Jun 2023 - 1:33

score 74 · Answer 8 · 2023-06-07 22:02:14

74

Views

Monotone convergence of $\sum\nolimits_{n=1}^\infty n^{- \alpha} f(nx) < \infty $

Published on 07 Jun 2023 - 22:02

score 485 · Answer 9 · 2023-06-12 23:22:34

485

Views

Can the supremum of an uncountable family of measures be replaced by the supremum over a countable subfamily?

Published on 12 Jun 2023 - 23:22

score 51 · Answer 10 · 2023-06-13 13:12:05

Prove that there exists a sequence of uniformly bounded measurable function such that ${f_n(x)}$ does not converge uniformly on any unbounded set

score 105 · Answer 11 · 2023-06-14 06:06:18

105

Views

$f_n\to f$ a.u. on $X$ iff there is $\{\varepsilon_n\}\downarrow 0$ s.t. $\lim_{k\to\infty}\mu(\bigcup_{n\geq k}\{|f_n-f|\geq\varepsilon_n\})=0$

Published on 14 Jun 2023 - 6:06

score 36 · Answer 12 · 2023-06-14 16:10:09

36

Views

Understanding part of a proof of dominated convergence Theorem from folland

Published on 14 Jun 2023 - 16:10

score 76 · Answer 13 · 2026-03-26 04:34:41

76

Views

Density of Sequences of Hilbert Spaces and Bochner Spaces

Published on 26 Mar 2026 - 4:34

score 48 · Answer 14 · 2026-03-26 11:00:07

48

Views

Measurability with respect to complete filtration

Published on 26 Mar 2026 - 11:00

score 29 · Answer 15 · 2026-03-25 12:51:59

29

Views

$f: E \rightarrow \mathbb{R}$ is $\mathcal{E}$-measurable if and only if $f^{+}$ and $f^{-}$ are $\mathcal{E}$-measurable

Published on 25 Mar 2026 - 12:51

List Question

Show that $f_n\overset{L_p}{\longrightarrow}f\Rightarrow f_n\overset{\mu}{\longrightarrow}f$

Show that $P_n\Rightarrow P$ if and only if $P_n(x_k)\rightarrow P(x_k)$ for every $k\ge$ and $\sum_{k=1}^\infty P(x_k)=1$

Elementary functions are measurable

Right-continuous, real valued functions of a real variable are $\mathcal{B}$-measurable

Prove that $\sup\limits_nf_n$ and $\inf\limits_nf_n$ are measurable.

Proving lower bound for an integral

How to understand that a function is measurable for a.e. x?

Monotone convergence of $\sum\nolimits_{n=1}^\infty n^{- \alpha} f(nx) < \infty $

Can the supremum of an uncountable family of measures be replaced by the supremum over a countable subfamily?

Prove that there exists a sequence of uniformly bounded measurable function such that ${f_n(x)}$ does not converge uniformly on any unbounded set

$f_n\to f$ a.u. on $X$ iff there is $\{\varepsilon_n\}\downarrow 0$ s.t. $\lim_{k\to\infty}\mu(\bigcup_{n\geq k}\{|f_n-f|\geq\varepsilon_n\})=0$

Understanding part of a proof of dominated convergence Theorem from folland

Density of Sequences of Hilbert Spaces and Bochner Spaces

Measurability with respect to complete filtration

$f: E \rightarrow \mathbb{R}$ is $\mathcal{E}$-measurable if and only if $f^{+}$ and $f^{-}$ are $\mathcal{E}$-measurable

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