Question

Measure decomposition

score 302 · Answer 1 · 2018-06-09 13:32:43

302

Views

Measure decomposition

Published on 09 Jun 2018 - 13:32

score 879 · Answer 2 · 2018-06-10 14:10:31

879

Views

Is it $\mu(dx)$ or $d\mu(x)$? or they are equal?

Published on 10 Jun 2018 - 14:10

score 34 · Answer 3 · 2018-06-11 10:13:07

34

Views

measurability of function that differs from continuous function

Published on 11 Jun 2018 - 10:13

score 175 · Answer 4 · 2026-03-26 04:35:00

175

Views

If $fg_n$ converges in $L^p$ for all $f\in L^p$, then $g_n$ converges in $L^\infty$.

Published on 26 Mar 2026 - 4:35

score 49 · Answer 5 · 2026-03-26 04:35:04

49

Views

$ \varphi(x_1, \dots , x_n) = (c_1x_1, \dots, c_nx_n)$. Then, $\mu(\varphi(A)) = |c_1\dots c_n|\mu(A)$, where $\mu$ is the Lebesgue measure

Published on 26 Mar 2026 - 4:35

score 26 · Answer 6 · 2018-06-12 15:28:56

26

Views

$f(x) \in \mathcal{M} (\mathbb{R}, \mathfrak{B}) \Leftrightarrow g_y(x):=f(x+y) \in \mathcal{M} (\mathbb{R}, \mathfrak{B})$

Published on 12 Jun 2018 - 15:28

score 498 · Answer 7 · 2018-06-14 21:01:14

498

Views

Lebesgue Measure and Isometries on the Unit Square

Published on 14 Jun 2018 - 21:01

score 70 · Answer 8 · 2026-03-26 07:37:30

70

Views

Lebesgue integrals

Published on 26 Mar 2026 - 7:37

score 108 · Answer 9 · 2026-03-26 04:35:12

108

Views

How do we know that the point of discontinuities of $f$ can be covered by at most countable many rectangles?

Published on 26 Mar 2026 - 4:35

score 370 · Answer 10 · 2018-06-16 13:10:49

370

Views

Partition of the unit interval into totally disconnected sets with positive measure

Published on 16 Jun 2018 - 13:10

score 86 · Answer 11 · 2018-06-17 08:55:41

86

Views

Suppose $A \cup B$ is measurable then prove the following property with respect to Outer and Inner measure

Published on 17 Jun 2018 - 8:55

score 51 · Answer 12 · 2026-03-25 01:14:16

51

Views

Interesting nonmeasurable function, need help understanding

Published on 25 Mar 2026 - 1:14

score 587 · Answer 13 · 2018-06-17 22:04:42

587

Views

Show $\operatorname{Char}_{E}(x)$ is measurable iff $E$ is measurable

Published on 17 Jun 2018 - 22:04

score 1.7k · Answer 14 · 2018-06-17 22:26:15

1.7k

Views

Showing function is measurable iff it's measurable for each rational

Published on 17 Jun 2018 - 22:26

score 128 · Answer 15 · 2018-06-17 22:53:53

128

Views

$f$ is measurable, $B$ is a Borel set $\rightarrow f^{-1}(B)$ is measurable.

Published on 17 Jun 2018 - 22:53

List Question

Measure decomposition

Is it $\mu(dx)$ or $d\mu(x)$? or they are equal?

measurability of function that differs from continuous function

If $fg_n$ converges in $L^p$ for all $f\in L^p$, then $g_n$ converges in $L^\infty$.

$ \varphi(x_1, \dots , x_n) = (c_1x_1, \dots, c_nx_n)$. Then, $\mu(\varphi(A)) = |c_1\dots c_n|\mu(A)$, where $\mu$ is the Lebesgue measure

$f(x) \in \mathcal{M} (\mathbb{R}, \mathfrak{B}) \Leftrightarrow g_y(x):=f(x+y) \in \mathcal{M} (\mathbb{R}, \mathfrak{B})$

Lebesgue Measure and Isometries on the Unit Square

Lebesgue integrals

How do we know that the point of discontinuities of $f$ can be covered by at most countable many rectangles?

Partition of the unit interval into totally disconnected sets with positive measure

Suppose $A \cup B$ is measurable then prove the following property with respect to Outer and Inner measure

Interesting nonmeasurable function, need help understanding

Show $\operatorname{Char}_{E}(x)$ is measurable iff $E$ is measurable

Showing function is measurable iff it's measurable for each rational

$f$ is measurable, $B$ is a Borel set $\rightarrow f^{-1}(B)$ is measurable.

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