Question

Why is $L^\infty \subset L^1$?

score 1k · Answer 1 · 2021-09-26 09:50:37

1k

Views

Why is $L^\infty \subset L^1$?

Published on 26 Sep 2021 - 9:50

score 126 · Answer 2 · 2021-09-26 14:56:41

126

Views

How to prove this function is measurable?

Published on 26 Sep 2021 - 14:56

score 806 · Answer 3 · 2021-09-26 19:34:55

806

Views

Prove that "pointwise maximum and minimum of two measurable functions are measurable."

Published on 26 Sep 2021 - 19:34

score 955 · Answer 4 · 2021-09-29 23:26:49

955

Views

Prove that $L^p(\Omega) \subset L^q (\Omega)$ whenever $\Omega$ is a bounded open set and $1 ≤ q ≤ p ≤ +\infty$.

Published on 29 Sep 2021 - 23:26

score 83 · Answer 5 · 2021-09-30 22:15:52

83

Views

Prove that the integral of a nonnegative lebesgue measurable function h is the integral of measures m({h>a} over [0,\infty).

Published on 30 Sep 2021 - 22:15

score 101 · Answer 6 · 2021-09-30 23:02:59

101

Views

The integral of a nonegative Lebesgue measurable function is equivalent to the following.

Published on 30 Sep 2021 - 23:02

score 93 · Answer 7 · 2021-10-05 16:15:40

93

Views

Show that $\left \{ g\geq a\right \} \in \mathcal{B}(\mathbb{R})$ for all $a\in \mathbb{R}$ and that $\mathcal{M}^+(\mathcal{B}(\mathbb{R}))$

Published on 05 Oct 2021 - 16:15

score 112 · Answer 8 · 2021-10-06 19:43:24

112

Views

Measurable w.r.t a set but not w.r.t another set

Published on 06 Oct 2021 - 19:43

score 77 · Answer 9 · 2021-10-07 00:18:24

77

Views

Extension of Dirac measure

Published on 07 Oct 2021 - 0:18

score 229 · Answer 10 · 2021-10-09 13:52:45

229

Views

Example of non measurable stochastic process

Published on 09 Oct 2021 - 13:52

score 43 · Answer 11 · 2021-10-10 17:39:32

43

Views

Proving a Function $X \to \mathbb{R}$ Measurable

Published on 10 Oct 2021 - 17:39

score 33 · Answer 12 · 2021-10-11 13:46:29

33

Views

Show that $\mu_{f+g}(\lambda_1+\lambda_2) \le \mu_f(\lambda_1)+\mu_g(\lambda_2)$ for $\lambda_1, \lambda_2 \ge 0$

Published on 11 Oct 2021 - 13:46

score 179 · Answer 13 · 2021-10-11 23:57:39

179

Views

Necessary topological conditions for all Caratheodory functions to be measurable

Published on 11 Oct 2021 - 23:57

score 97 · Answer 14 · 2021-10-12 13:58:38

97

Views

Show that $\mu(\{x \in X \colon f(x)> \alpha\}) \leq \frac{1}{1 + \alpha^2}$ for all $\alpha > 0 $

Published on 12 Oct 2021 - 13:58

score 52 · Answer 15 · 2026-02-26 02:57:58

52

Views

If $ \lim_{k\rightarrow+\infty}\int_E \dfrac{f_k(x)}{1+f_k(x)}\mathrm{d}x=0, $ prove that $f_k$ converge to $0$ in measure.

Published on 26 Feb 2026 - 2:57

List Question

Why is $L^\infty \subset L^1$?

How to prove this function is measurable?

Prove that "pointwise maximum and minimum of two measurable functions are measurable."

Prove that $L^p(\Omega) \subset L^q (\Omega)$ whenever $\Omega$ is a bounded open set and $1 ≤ q ≤ p ≤ +\infty$.

Prove that the integral of a nonnegative lebesgue measurable function h is the integral of measures m({h>a} over [0,\infty).

The integral of a nonegative Lebesgue measurable function is equivalent to the following.

Show that $\left \{ g\geq a\right \} \in \mathcal{B}(\mathbb{R})$ for all $a\in \mathbb{R}$ and that $\mathcal{M}^+(\mathcal{B}(\mathbb{R}))$

Measurable w.r.t a set but not w.r.t another set

Extension of Dirac measure

Example of non measurable stochastic process

Proving a Function $X \to \mathbb{R}$ Measurable

Show that $\mu_{f+g}(\lambda_1+\lambda_2) \le \mu_f(\lambda_1)+\mu_g(\lambda_2)$ for $\lambda_1, \lambda_2 \ge 0$

Necessary topological conditions for all Caratheodory functions to be measurable

Show that $\mu(\{x \in X \colon f(x)> \alpha\}) \leq \frac{1}{1 + \alpha^2}$ for all $\alpha > 0 $

If $ \lim_{k\rightarrow+\infty}\int_E \dfrac{f_k(x)}{1+f_k(x)}\mathrm{d}x=0, $ prove that $f_k$ converge to $0$ in measure.

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