Question

Let $(f_n)$ be a Cauchy sequence in $S (X)$. There is a subsequence $(f_{n_k})$ and $f \in L^0(X)$ such that $f_{n_k} \to f$ a.e.

score 38 · Answer 1 · 2023-08-16 10:31:00

38

Views

Let $(f_n)$ be a Cauchy sequence in $S (X)$. There is a subsequence $(f_{n_k})$ and $f \in L^0(X)$ such that $f_{n_k} \to f$ a.e.

Published on 16 Aug 2023 - 10:31

score 53 · Answer 2 · 2026-03-27 16:27:48

53

Views

The metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions is complete

Published on 27 Mar 2026 - 16:27

score 33 · Answer 3 · 2023-08-16 11:31:55

33

Views

A characterization of convergence in the metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions

Published on 16 Aug 2023 - 11:31

score 43 · Answer 4 · 2023-08-16 11:54:03

43

Views

Let $f \in L^0(X \times Y)$. Then for $\mu$-a.e. $x \in X$ we have $f(x, \cdot) \in L^0(Y)$

Published on 16 Aug 2023 - 11:54

score 29 · Answer 5 · 2026-02-23 03:05:18

29

Views

Measurable Riemann Mapping Theorem on a simply connected set

Published on 23 Feb 2026 - 3:05

score 53 · Answer 6 · 2023-08-17 08:52:16

53

Views

Convergence in a complete measure is metrizable

Published on 17 Aug 2023 - 8:52

score 41 · Answer 7 · 2026-03-27 16:28:11

41

Views

The metric $\hat \rho (f, g) := \inf_{\delta >0} \{ \mu (|f - g| > \delta) +\delta \}$ on the space of $\mu$-measurable functions is complete

Published on 27 Mar 2026 - 16:28

score 37 · Answer 8 · 2023-08-18 14:42:01

37

Views

If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$, then $(f_n)$ is Cauchy in $(L^0 (Z), \rho_Z)$

Published on 18 Aug 2023 - 14:42

score 111 · Answer 9 · 2023-08-19 17:54:52

111

Views

Show that $T$ is not continuous when $p = \infty.$

Published on 19 Aug 2023 - 17:54

score 34 · Answer 10 · 2023-08-22 08:09:21

34

Views

Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Then the map $x \mapsto \|f(x, \cdot)\|_{L^p_{\text{loc}}}$ is measurable

Published on 22 Aug 2023 - 8:09

score 28 · Answer 11 · 2023-08-22 17:40:17

28

Views

If $f \in L^0 (X, L^p_{\text{loc}} (Y))$, then $f \in L^0 (Z)$

Published on 22 Aug 2023 - 17:40

score 29 · Answer 12 · 2023-08-23 15:01:25

29

Views

How can i get that $\mathbb{E}[(\mathbb{E}(X|Y)-g(Y))(X-\mathbb{E}(X|Y))]=0$?

Published on 23 Aug 2023 - 15:01

score 23 · Answer 13 · 2026-02-24 18:55:43

23

Views

Convergence in a complete measure implies a.e. convergence for a subsequence

Published on 24 Feb 2026 - 18:55

score 22 · Answer 14 · 2023-08-23 16:57:24

22

Views

If $\nu(Y) < \infty$ then $F: X \to L^0(Y), x \mapsto f(x, \cdot)$ is $\mu$-measurable

Published on 23 Aug 2023 - 16:57

score 19 · Answer 15 · 2023-08-23 17:50:17

19

Views

If $\| g_n \|_{L^0(X)} \to 0$ then $\| f_n \|_{L^0 (Z)} \to 0$

Published on 23 Aug 2023 - 17:50

List Question

Let $(f_n)$ be a Cauchy sequence in $S (X)$. There is a subsequence $(f_{n_k})$ and $f \in L^0(X)$ such that $f_{n_k} \to f$ a.e.

The metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions is complete

A characterization of convergence in the metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions

Let $f \in L^0(X \times Y)$. Then for $\mu$-a.e. $x \in X$ we have $f(x, \cdot) \in L^0(Y)$

Measurable Riemann Mapping Theorem on a simply connected set

Convergence in a complete measure is metrizable

The metric $\hat \rho (f, g) := \inf_{\delta >0} \{ \mu (|f - g| > \delta) +\delta \}$ on the space of $\mu$-measurable functions is complete

If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$, then $(f_n)$ is Cauchy in $(L^0 (Z), \rho_Z)$

Show that $T$ is not continuous when $p = \infty.$

Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Then the map $x \mapsto \|f(x, \cdot)\|_{L^p_{\text{loc}}}$ is measurable

If $f \in L^0 (X, L^p_{\text{loc}} (Y))$, then $f \in L^0 (Z)$

How can i get that $\mathbb{E}[(\mathbb{E}(X|Y)-g(Y))(X-\mathbb{E}(X|Y))]=0$?

Convergence in a complete measure implies a.e. convergence for a subsequence

If $\nu(Y) < \infty$ then $F: X \to L^0(Y), x \mapsto f(x, \cdot)$ is $\mu$-measurable

If $\| g_n \|_{L^0(X)} \to 0$ then $\| f_n \|_{L^0 (Z)} \to 0$

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