Question

Prob. 8, Chap. 2, in Royden's REAL ANALYSIS: The collection of finitely many open intervals covering the rational numbers in $[0, 1]$ ...

score 75 · Answer 1 · 2021-03-31 18:36:05

75

Views

Prob. 8, Chap. 2, in Royden's REAL ANALYSIS: The collection of finitely many open intervals covering the rational numbers in $[0, 1]$ ...

Published on 31 Mar 2021 - 18:36

score 574 · Answer 2 · 2021-04-01 15:22:41

574

Views

Range of measure function cannot be half open interval

Published on 01 Apr 2021 - 15:22

score 104 · Answer 3 · 2021-04-01 18:16:52

104

Views

$A,B\subset\mathbb{R}, |A|<\infty\Rightarrow |B-A|\geq |B|-|A|,\ |\cdot|$ outer measure

Published on 01 Apr 2021 - 18:16

score 38 · Answer 4 · 2021-04-02 11:42:25

38

Views

$A,\ B\subseteq R$ and $m(A)<\infty$, show that $m^(B\cap A^c)\geq m^(B)-m^(A)$

Published on 02 Apr 2021 - 11:42

score 159 · Answer 5 · 2021-04-02 13:32:36

159

Views

How to prove this set as a null set?

Published on 02 Apr 2021 - 13:32

score 947 · Answer 6 · 2021-04-03 13:31:00

947

Views

$A$ is Lebesgue measurable iff $\forall \epsilon > 0$, there exists open set $G$ such that $|G \setminus A| + |A \setminus G| < \epsilon$

Published on 03 Apr 2021 - 13:31

score 859 · Answer 7 · 2021-04-03 16:51:38

859

Views

Prob. 18, Chap. 2, in Royden's REAL ANALYSIS: If $E$ has finite outer measure, then there is an $F_\sigma$-set $F$ and a $G_\delta$-set $G$ with ...

Published on 03 Apr 2021 - 16:51

score 85 · Answer 8 · 2026-03-26 09:42:38

85

Views

Prob. 20, Chap. 2, in Royden's REAL ANALYSIS: Does $E$ necessarily have to have finite outer measure?

Published on 26 Mar 2026 - 9:42

score 141 · Answer 9 · 2021-04-04 11:19:03

141

Views

Prob. 22, Chap. 2, in Royden's REAL ANALYSIS: $m^{**}(A) = m^*(A)$

Published on 04 Apr 2021 - 11:19

score 45 · Answer 10 · 2021-04-12 10:40:14

45

Views

$X$ is locally compact, $\sigma$-compact, $T_2$. $\mu$ be a borel measure which is outer and finite on compact sets. Then $\mu$ is inner measure.

Published on 12 Apr 2021 - 10:40

score 224 · Answer 11 · 2021-04-12 14:32:52

224

Views

Regular outer measure resricted to a measurable subset

Published on 12 Apr 2021 - 14:32

score 175 · Answer 12 · 2021-04-13 00:09:10

175

Views

About some proofs using epsilon notation.

Published on 13 Apr 2021 - 0:09

score 139 · Answer 13 · 2021-04-13 08:54:21

139

Views

An example for $m^{}(\bigcap_{k=1}^{\infty} A_k) \ne \lim_{k \to \infty} m^{}(A_k)$?

Published on 13 Apr 2021 - 8:54

score 131 · Answer 14 · 2021-04-13 09:14:29

131

Views

Show that for any $E=E_1 \cup E_2$, $μ(E)=μ^∗(E_1)=μ^∗(E_2)$

Published on 13 Apr 2021 - 9:14

score 172 · Answer 15 · 2026-03-26 09:42:28

172

Views

Show that $\inf {\{m^∗(A \cap B): A, B \ m^∗ \text{–measurable}, A \supseteq E, B \supseteq X\backslash E}\} > 0$.

Published on 26 Mar 2026 - 9:42

List Question

Prob. 8, Chap. 2, in Royden's REAL ANALYSIS: The collection of finitely many open intervals covering the rational numbers in $[0, 1]$ ...

Range of measure function cannot be half open interval

$A,B\subset\mathbb{R}, |A|<\infty\Rightarrow |B-A|\geq |B|-|A|,\ |\cdot|$ outer measure

$A,\ B\subseteq R$ and $m(A)<\infty$, show that $m^(B\cap A^c)\geq m^(B)-m^(A)$

How to prove this set as a null set?

$A$ is Lebesgue measurable iff $\forall \epsilon > 0$, there exists open set $G$ such that $|G \setminus A| + |A \setminus G| < \epsilon$

Prob. 18, Chap. 2, in Royden's REAL ANALYSIS: If $E$ has finite outer measure, then there is an $F_\sigma$-set $F$ and a $G_\delta$-set $G$ with ...

Prob. 20, Chap. 2, in Royden's REAL ANALYSIS: Does $E$ necessarily have to have finite outer measure?

Prob. 22, Chap. 2, in Royden's REAL ANALYSIS: $m^{**}(A) = m^*(A)$

$X$ is locally compact, $\sigma$-compact, $T_2$. $\mu$ be a borel measure which is outer and finite on compact sets. Then $\mu$ is inner measure.

Regular outer measure resricted to a measurable subset

About some proofs using epsilon notation.

An example for $m^{}(\bigcap_{k=1}^{\infty} A_k) \ne \lim_{k \to \infty} m^{}(A_k)$?

Show that for any $E=E_1 \cup E_2$, $μ(E)=μ^∗(E_1)=μ^∗(E_2)$

Show that $\inf {\{m^∗(A \cap B): A, B \ m^∗ \text{–measurable}, A \supseteq E, B \supseteq X\backslash E}\} > 0$.

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