Question

Prove that a set $E \subset R$ has non-measurable subsets if and only if $m∗(E)>0$

score 346 · Answer 1 · 2021-05-20 17:07:31

346

Views

Prove that a set $E \subset R$ has non-measurable subsets if and only if $m∗(E)>0$

Published on 20 May 2021 - 17:07

score 47 · Answer 2 · 2021-05-23 17:09:06

47

Views

Prove $\lim_{n \to \infty}\mu(A_n) = 0$

Published on 23 May 2021 - 17:09

score 40 · Answer 3 · 2021-05-24 06:52:50

40

Views

An algebra is a $\sigma$-algebra if for all $E_1 \subset E_2 \subset\cdots$, $E_k \in A$, one has $\cup E_k \in A$

Published on 24 May 2021 - 6:52

score 121 · Answer 4 · 2021-06-02 04:52:46

121

Views

Prob. 24, Chap. 2, in Royden's REAL ANALYSIS: $m\left(E_1\cup E_2\right)+m\left(E_1\cap E_2\right)=m\left(E_1\right)+m\left(E_2\right)$

Published on 02 Jun 2021 - 4:52

score 49 · Answer 5 · 2021-06-03 20:16:18

49

Views

If $A_n=\{x\in \mathbb{R}:f(x)<\frac{-1}{n}\}$ are measurable, prove that $A=\{x\in \mathbb{R}:f(x)\le0\}$ is measurable too.

Published on 03 Jun 2021 - 20:16

score 255 · Answer 6 · 2021-06-11 08:22:00

255

Views

Prob. 25, Chap. 2, in Royden's REAL ANALYSIS: The assumption that $m\left(B_1\right)<\infty$ is necessary in ...

Published on 11 Jun 2021 - 8:22

score 147 · Answer 7 · 2021-06-11 09:13:23

147

Views

Measurability of Hausdorff measure under bi-Lipschitz mappings

Published on 11 Jun 2021 - 9:13

score 62 · Answer 8 · 2021-06-12 05:34:37

62

Views

If non negative functions $f,g : \mathbb{R^n} \to [0, \infty]$ are measurable, then $f+g$ is measurable.

Published on 12 Jun 2021 - 5:34

score 41 · Answer 9 · 2021-06-12 11:07:13

41

Views

Proof of the fact that $\{ x \in \mathbb{R^n} | f(x)\neq g(x) \}$ is lebesgue-measurable.

Published on 12 Jun 2021 - 11:07

score 237 · Answer 10 · 2021-06-12 15:26:32

237

Views

$f,g$ are lebesgue-measurable and finite a.e. Then, prove that $f\cdot g$ is lebesgue-measurable.

Published on 12 Jun 2021 - 15:26

score 84 · Answer 11 · 2021-06-21 04:04:42

84

Views

Is $f \equiv \infty$ Lebesgue measurable?

Published on 21 Jun 2021 - 4:04

score 63 · Answer 12 · 2021-06-25 22:18:57

63

Views

A measure $\mu$ on $X$ is $\sigma$-finite if and only if the constant function $1_{X}$ is strongly $\mu$-measurable

Published on 25 Jun 2021 - 22:18

score 853 · Answer 13 · 2021-07-07 00:34:57

853

Views

Sequence of sets converge to zero in measure if integral converges to zero.

Published on 07 Jul 2021 - 0:34

score 228 · Answer 14 · 2021-07-08 15:31:40

228

Views

In real analysis, is constant function measurable??

Published on 08 Jul 2021 - 15:31

score 45 · Answer 15 · 2021-07-11 12:01:04

45

Views

Common points of two measurable functions

Published on 11 Jul 2021 - 12:01

List Question

Prove that a set $E \subset R$ has non-measurable subsets if and only if $m∗(E)>0$

Prove $\lim_{n \to \infty}\mu(A_n) = 0$

An algebra is a $\sigma$-algebra if for all $E_1 \subset E_2 \subset\cdots$, $E_k \in A$, one has $\cup E_k \in A$

Prob. 24, Chap. 2, in Royden's REAL ANALYSIS: $m\left(E_1\cup E_2\right)+m\left(E_1\cap E_2\right)=m\left(E_1\right)+m\left(E_2\right)$

If $A_n=\{x\in \mathbb{R}:f(x)<\frac{-1}{n}\}$ are measurable, prove that $A=\{x\in \mathbb{R}:f(x)\le0\}$ is measurable too.

Prob. 25, Chap. 2, in Royden's REAL ANALYSIS: The assumption that $m\left(B_1\right)<\infty$ is necessary in ...

Measurability of Hausdorff measure under bi-Lipschitz mappings

If non negative functions $f,g : \mathbb{R^n} \to [0, \infty]$ are measurable, then $f+g$ is measurable.

Proof of the fact that $\{ x \in \mathbb{R^n} | f(x)\neq g(x) \}$ is lebesgue-measurable.

$f,g$ are lebesgue-measurable and finite a.e. Then, prove that $f\cdot g$ is lebesgue-measurable.

Is $f \equiv \infty$ Lebesgue measurable?

A measure $\mu$ on $X$ is $\sigma$-finite if and only if the constant function $1_{X}$ is strongly $\mu$-measurable

Sequence of sets converge to zero in measure if integral converges to zero.

In real analysis, is constant function measurable??

Common points of two measurable functions

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