Question

Continuity of Measure for Decreasing Sequence

score 220 · Answer 1 · 2021-02-24 18:05:50

220

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Continuity of Measure for Decreasing Sequence

Published on 24 Feb 2021 - 18:05

score 154 · Answer 2 · 2021-03-02 09:12:16

154

Views

Why is the following a probability measure?

Published on 02 Mar 2021 - 9:12

score 96 · Answer 3 · 2021-03-08 02:46:37

96

Views

Show that $m^{}([a, b]\backslash G)=b − a- m^{}{(G)}$

Published on 08 Mar 2021 - 2:46

score 32 · Answer 4 · 2021-03-08 13:27:13

32

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Is the intersection of any subset of $X$ and a $\lambda$-measurable set $\lambda$-measurable?

Published on 08 Mar 2021 - 13:27

score 278 · Answer 5 · 2021-03-09 15:30:38

278

Views

For every measurable set, there is a set in the $\sigma$-algebra with the same outer measure

Published on 09 Mar 2021 - 15:30

score 924 · Answer 6 · 2021-03-10 13:42:06

924

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$A\subset\mathbb{R}, t>0\Rightarrow |A|=|A\cap (-t,t)|+|A\cap (\mathbb{R}-(-t,t))|$

Published on 10 Mar 2021 - 13:42

score 203 · Answer 7 · 2021-03-13 16:46:47

Let $\mu$ be a regular outer measure on $X$, $\mu(X) < \infty$. If $A \subseteq X $ is set such that $\mu(A)+\mu(A^c)=\mu(X)$ then $A$ is measurable

score 285 · Answer 8 · 2021-03-14 11:37:53

285

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Outer measure of union of two open intervals with empty intersection

Published on 14 Mar 2021 - 11:37

score 894 · Answer 9 · 2021-03-16 16:35:27

894

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$I_1,I_2,\dots$ disjoint sequence of open intervals $\Rightarrow |\bigcup_{k=1}^{\infty} I_k|=\sum_{k=1}^{\infty}\ell(I_k)$

Published on 16 Mar 2021 - 16:35

score 584 · Answer 10 · 2021-03-19 19:38:04

584

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$|A|=\lim_{t\to\infty} |A\cap (-t,t)|,\ A\subset\mathbb{R}, t>0$

Published on 19 Mar 2021 - 19:38

score 457 · Answer 11 · 2021-03-20 12:39:14

457

Views

$\forall\varepsilon>0$ there exists a subset $F$ of $[0,1]$ that is closed, contains only irrational numbers and $|F|>1-\varepsilon$

Published on 20 Mar 2021 - 12:39

score 85 · Answer 12 · 2021-03-23 05:19:32

85

Views

For a regular Borel measure with infinite support, how to find a decreasing sequence of positive measure open sets?

Published on 23 Mar 2021 - 5:19

score 1k · Answer 13 · 2021-03-29 14:11:50

1k

Views

$A\subset\mathbb{R}, t\in\mathbb{R}\Rightarrow |tA|=|t||A|$

Published on 29 Mar 2021 - 14:11

score 629 · Answer 14 · 2021-03-30 10:53:08

629

Views

Outer measure of intervals: $|(a,b)|=|[a,b)|=|(a,b]|=b-a$

Published on 30 Mar 2021 - 10:53

score 324 · Answer 15 · 2021-03-31 12:13:11

324

Views

Prob. 23, Chap. 2, in Royden's REAL ANALYSIS: How is $m^{***}(A)=\sup\{m^*(F) | F \subset A, F \mbox{ closed} \}$ related to the outer measure?

Published on 31 Mar 2021 - 12:13

List Question

Continuity of Measure for Decreasing Sequence

Why is the following a probability measure?

Show that $m^{}([a, b]\backslash G)=b − a- m^{}{(G)}$

Is the intersection of any subset of $X$ and a $\lambda$-measurable set $\lambda$-measurable?

For every measurable set, there is a set in the $\sigma$-algebra with the same outer measure

$A\subset\mathbb{R}, t>0\Rightarrow |A|=|A\cap (-t,t)|+|A\cap (\mathbb{R}-(-t,t))|$

Let $\mu$ be a regular outer measure on $X$, $\mu(X) < \infty$. If $A \subseteq X $ is set such that $\mu(A)+\mu(A^c)=\mu(X)$ then $A$ is measurable

Outer measure of union of two open intervals with empty intersection

$I_1,I_2,\dots$ disjoint sequence of open intervals $\Rightarrow |\bigcup_{k=1}^{\infty} I_k|=\sum_{k=1}^{\infty}\ell(I_k)$

$|A|=\lim_{t\to\infty} |A\cap (-t,t)|,\ A\subset\mathbb{R}, t>0$

$\forall\varepsilon>0$ there exists a subset $F$ of $[0,1]$ that is closed, contains only irrational numbers and $|F|>1-\varepsilon$

For a regular Borel measure with infinite support, how to find a decreasing sequence of positive measure open sets?

$A\subset\mathbb{R}, t\in\mathbb{R}\Rightarrow |tA|=|t||A|$

Outer measure of intervals: $|(a,b)|=|[a,b)|=|(a,b]|=b-a$

Prob. 23, Chap. 2, in Royden's REAL ANALYSIS: How is $m^{***}(A)=\sup\{m^*(F) | F \subset A, F \mbox{ closed} \}$ related to the outer measure?

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