Question

If $f$ is pointwise integrable, does it hold $\int Lf\:{\rm d}\mu=L\int f\:{\rm d}\mu$ for every bounded linear operator $L$?

score 53 · Answer 1 · 2022-07-07 10:26:23

53

Views

If $f$ is pointwise integrable, does it hold $\int Lf\:{\rm d}\mu=L\int f\:{\rm d}\mu$ for every bounded linear operator $L$?

Published on 07 Jul 2022 - 10:26

score 145 · Answer 2 · 2022-08-04 09:57:22

145

Views

Let $E$ be a Banach space and $p \in (1, \infty)$. Is $L_{p}(X, \mu, E)$ uniformly convex?

Published on 04 Aug 2022 - 9:57

score 172 · Answer 3 · 2022-08-04 11:10:04

172

Views

Let $E$ be a Banach space, $p \in (1, \infty)$, and $L_p := L_{p}(X, \mu, E)$. Is $(L_p)^* \cong L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$?

Published on 04 Aug 2022 - 11:10

score 61 · Answer 4 · 2022-08-07 08:15:14

61

Views

Completing a measure space does not affect the strong measurability of a function

Published on 07 Aug 2022 - 8:15

score 40 · Answer 5 · 2022-08-08 15:37:40

40

Views

Completing a measure space does not affect the integrability of a function

Published on 08 Aug 2022 - 15:37

score 119 · Answer 6 · 2022-08-09 02:55:15

119

Views

Lebesgue integral coincides with Bochner integral if the former is finite

Published on 09 Aug 2022 - 2:55

score 139 · Answer 7 · 2022-08-09 08:26:59

139

Views

How does professor Amann conclude that "Corollary 1.5 remains true for incomplete measure spaces"?

Published on 09 Aug 2022 - 8:26

score 100 · Answer 8 · 2022-08-14 02:15:06

100

Views

If $f_n \to f$ $\mu$-a.e. and $\|f_n\|_1 \to \|f\|$, then $\|f_n - f\| \to 0$

Published on 14 Aug 2022 - 2:15

score 71 · Answer 9 · 2022-09-09 22:39:57

71

Views

Integral in lemma 1.20 in Takesaki's book

Published on 09 Sep 2022 - 22:39

score 259 · Answer 10 · 2022-10-05 22:27:45

259

Views

Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?

Published on 05 Oct 2022 - 22:27

score 72 · Answer 11 · 2022-10-07 09:29:32

72

Views

If $\int_A f \mathrm d \mu \in F$ for every measurable set $A$, then $f$ takes values on $F$ almost surely

Published on 07 Oct 2022 - 9:29

score 78 · Answer 12 · 2022-10-07 19:52:14

78

Views

Generalize Theorem 1.40 in Rudin's Real and Complex Analysis

Published on 07 Oct 2022 - 19:52

score 69 · Answer 13 · 2026-03-26 06:13:56

69

Views

Pushforward measure: change-of-variables formula for Banach spaces

Published on 26 Mar 2026 - 6:13

score 37 · Answer 14 · 2022-11-09 14:40:06

37

Views

If $\left | \frac{1}{\mu(A)} \int_A f \mathrm d \mu \right | \in G$ for all $A \in \mathcal A$ then $|f(x)| \in G$ for almost all $x \in X$

Published on 09 Nov 2022 - 14:40

score 60 · Answer 15 · 2022-11-10 00:42:06

60

Views

If $X$ is locally compact separable, then $\mathcal C_c(X)$ is dense in $\big (\mathcal L_p, \|\cdot\|_{\mathcal L_p} \big)$ for all $p\in [1,\infty)$

Published on 10 Nov 2022 - 0:42

List Question

If $f$ is pointwise integrable, does it hold $\int Lf\:{\rm d}\mu=L\int f\:{\rm d}\mu$ for every bounded linear operator $L$?

Let $E$ be a Banach space and $p \in (1, \infty)$. Is $L_{p}(X, \mu, E)$ uniformly convex?

Let $E$ be a Banach space, $p \in (1, \infty)$, and $L_p := L_{p}(X, \mu, E)$. Is $(L_p)^* \cong L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$?

Completing a measure space does not affect the strong measurability of a function

Completing a measure space does not affect the integrability of a function

Lebesgue integral coincides with Bochner integral if the former is finite

How does professor Amann conclude that "Corollary 1.5 remains true for incomplete measure spaces"?

If $f_n \to f$ $\mu$-a.e. and $\|f_n\|_1 \to \|f\|$, then $\|f_n - f\| \to 0$

Integral in lemma 1.20 in Takesaki's book

Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?

If $\int_A f \mathrm d \mu \in F$ for every measurable set $A$, then $f$ takes values on $F$ almost surely

Generalize Theorem 1.40 in Rudin's Real and Complex Analysis

Pushforward measure: change-of-variables formula for Banach spaces

If $\left | \frac{1}{\mu(A)} \int_A f \mathrm d \mu \right | \in G$ for all $A \in \mathcal A$ then $|f(x)| \in G$ for almost all $x \in X$

If $X$ is locally compact separable, then $\mathcal C_c(X)$ is dense in $\big (\mathcal L_p, \|\cdot\|_{\mathcal L_p} \big)$ for all $p\in [1,\infty)$

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