Question

if $E\subset\mathbb{R}$ is Lebesgue measurable, $m(E)<\infty$,$F\subset E$, then $m(E)=m^(F)+m^(E\setminus F)$ if and only if $F$ is measurable.

score 143 · Answer 1 · 2026-04-17 05:58:59

143

Views

if $E\subset\mathbb{R}$ is Lebesgue measurable, $m(E)<\infty$,$F\subset E$, then $m(E)=m^(F)+m^(E\setminus F)$ if and only if $F$ is measurable.

Published on 17 Apr 2026 - 5:58

score 186 · Answer 2 · 2026-04-11 13:27:08

186

Views

Are closure of open subsets of [0, 1] “almost open”?

Published on 11 Apr 2026 - 13:27

score 57 · Answer 3 · 2026-04-17 09:20:54

57

Views

If $E$ is Lebesgue measurable and non-empty, but $A$ is not measurable, then $E\times A$ is not measurable in $\mathbb{R^2}$?

Published on 17 Apr 2026 - 9:20

score 34 · Answer 4 · 2026-04-17 07:29:54

34

Views

An example of a measurable function $f$ such that both $f_{\pm}$ have infinite integral on each interval

Published on 17 Apr 2026 - 7:29

score 231 · Answer 5 · 2026-04-16 03:21:45

231

Views

$\exists\, f\in L^2(X)$ s.t. $\{gf\mid g\in C(X, \mathbb{C})\} \subset L^2(X)$ is dense

Published on 16 Apr 2026 - 3:21

score 39 · Answer 6 · 2026-04-12 19:08:41

39

Views

Reducing integrals over abstract spaces to integrals on $\mathbb{R}_{+}$ with respect to the Lebesgue measure

Published on 12 Apr 2026 - 19:08

score 88 · Answer 7 · 2026-04-17 10:10:32

88

Views

One-sided Hardy-Littlewood inequality for monotone function

Published on 17 Apr 2026 - 10:10

score 123 · Answer 8 · 2026-04-16 12:57:53

123

Views

$f:[0,\infty)\to \mathbb{R}$ right continuous with left limits. Then is $\{y:\lambda(t:f(t)=y)=0\}$ dense in $\mathbb{R}$?

Published on 16 Apr 2026 - 12:57

score 46 · Answer 9 · 2026-04-09 10:38:09

46

Views

Monotone differentiation theorem

Published on 09 Apr 2026 - 10:38

score 35 · Answer 10 · 2026-04-12 23:10:20

35

Views

Measure of the set $Z \times N$, where $Z$ has zero measure and $N$ is non-measurable

Published on 12 Apr 2026 - 23:10

score 74 · Answer 11 · 2026-04-16 11:11:03

74

Views

Comparing $L^\infty$ norm of $f$ from $0$ to $s$ to $f(s)$.

Published on 16 Apr 2026 - 11:11

score 79 · Answer 12 · 2026-04-15 19:56:42

79

Views

Prove that for each $\varepsilon > 0$, there exists $M>0$ such that $m^* (E \setminus [-M,M]) < \varepsilon$

Published on 15 Apr 2026 - 19:56

score 79 · Answer 13 · 2026-04-17 01:31:14

79

Views

Show that $\displaystyle\int_{\mathbb{R}} \lim \inf _{\epsilon \to 0} |f_\epsilon| \leq \displaystyle\int_{\mathbb{R}} |f|$

Published on 17 Apr 2026 - 1:31

score 39 · Answer 14 · 2026-04-17 11:56:27

39

Views

Prove that if either $f \in (L^p \cap C^m)(\mathbb{R}^n)$, or $g \in (L^q \cap C^m)(\mathbb{R}^n)$, then $f \ast g \in C^m$

Published on 17 Apr 2026 - 11:56

score 62 · Answer 15 · 2026-04-13 02:39:44

62

Views

Give a function in $L^p$ but not in weak $L^p$

Published on 13 Apr 2026 - 2:39

List Question

if $E\subset\mathbb{R}$ is Lebesgue measurable, $m(E)<\infty$,$F\subset E$, then $m(E)=m^(F)+m^(E\setminus F)$ if and only if $F$ is measurable.

Are closure of open subsets of [0, 1] “almost open”?

If $E$ is Lebesgue measurable and non-empty, but $A$ is not measurable, then $E\times A$ is not measurable in $\mathbb{R^2}$?

An example of a measurable function $f$ such that both $f_{\pm}$ have infinite integral on each interval

$\exists\, f\in L^2(X)$ s.t. $\{gf\mid g\in C(X, \mathbb{C})\} \subset L^2(X)$ is dense

Reducing integrals over abstract spaces to integrals on $\mathbb{R}_{+}$ with respect to the Lebesgue measure

One-sided Hardy-Littlewood inequality for monotone function

$f:[0,\infty)\to \mathbb{R}$ right continuous with left limits. Then is $\{y:\lambda(t:f(t)=y)=0\}$ dense in $\mathbb{R}$?

Monotone differentiation theorem

Measure of the set $Z \times N$, where $Z$ has zero measure and $N$ is non-measurable

Comparing $L^\infty$ norm of $f$ from $0$ to $s$ to $f(s)$.

Prove that for each $\varepsilon > 0$, there exists $M>0$ such that $m^* (E \setminus [-M,M]) < \varepsilon$

Show that $\displaystyle\int_{\mathbb{R}} \lim \inf _{\epsilon \to 0} |f_\epsilon| \leq \displaystyle\int_{\mathbb{R}} |f|$

Prove that if either $f \in (L^p \cap C^m)(\mathbb{R}^n)$, or $g \in (L^q \cap C^m)(\mathbb{R}^n)$, then $f \ast g \in C^m$

Give a function in $L^p$ but not in weak $L^p$

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