Question

Prove that if $f$ is nonnegative, measurable and $E_k \nearrow E$, then $\lim_k \int_{E_k} f = \int_E f$.

score 208 · Answer 1 · 2026-03-27 17:57:35

208

Views

Prove that if $f$ is nonnegative, measurable and $E_k \nearrow E$, then $\lim_k \int_{E_k} f = \int_E f$.

Published on 27 Mar 2026 - 17:57

score 284 · Answer 2 · 2026-03-27 17:52:55

284

Views

In a measurable partition of an interval, the sum of the measures of the subsets in the partition equals the length of the interval

Published on 27 Mar 2026 - 17:52

score 108 · Answer 3 · 2026-03-29 22:28:58

108

Views

Sets of null (Lebesgue)-measure and sigma compacts

Published on 29 Mar 2026 - 22:28

score 653 · Answer 4 · 2018-11-16 19:36:17

653

Views

Every subset of a subspace of $\mathbb{R}^n$ of dim $<n$ has measure 0

Published on 16 Nov 2018 - 19:36

score 54 · Answer 5 · 2018-11-16 23:21:08

54

Views

$f:[a,b] \rightarrow \Bbb R$ measurable function, $f\geq0$ a.e. If $f^{-1}((0,\infty))$ has measure $>0$ , $f^{-1}((1/n,\infty))$ does for some n.

Published on 16 Nov 2018 - 23:21

score 320 · Answer 6 · 2026-03-27 19:32:36

320

Views

Hardy Littlewood maximal function $|h(x)|<|h^*(x)|$ for almost every x

Published on 27 Mar 2026 - 19:32

score 115 · Answer 7 · 2018-11-17 21:36:38

115

Views

Caratheodory measure problem

Published on 17 Nov 2018 - 21:36

score 68 · Answer 8 · 2026-03-25 19:28:31

68

Views

Countable $\varepsilon$-net of measurable sets

Published on 25 Mar 2026 - 19:28

score 556 · Answer 9 · 2026-03-25 04:42:46

556

Views

Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra

Published on 25 Mar 2026 - 4:42

score 65 · Answer 10 · 2026-03-27 21:23:28

65

Views

A "nice result" in Functional Analysis

Published on 27 Mar 2026 - 21:23

score 235 · Answer 11 · 2026-03-27 21:22:57

235

Views

Bogachev change of variables theorem.

Published on 27 Mar 2026 - 21:22

score 74 · Answer 12 · 2018-11-18 16:42:01

74

Views

$\mu$, a Borel measure on $\mathbb{R}$ satisfies $\mu(K)<\infty$ for compact K, $\mu(x+E)=\mu(E)$. Then $\mu(E) =c\cdot\lambda(E)$

Published on 18 Nov 2018 - 16:42

score 295 · Answer 13 · 2026-03-27 21:17:33

295

Views

Prove that $\lim_{n\rightarrow \infty} \int_{[-n,n]} f\,d\lambda= \int f\,d\lambda.$

Published on 27 Mar 2026 - 21:17

score 105 · Answer 14 · 2026-03-27 21:18:10

105

Views

What is a tube in $\mathbb{R}^n$?

Published on 27 Mar 2026 - 21:18

score 360 · Answer 15 · 2026-03-27 21:17:40

360

Views

Lebesgue integration: integral of continuous function tends to infinity

Published on 27 Mar 2026 - 21:17

List Question

Prove that if $f$ is nonnegative, measurable and $E_k \nearrow E$, then $\lim_k \int_{E_k} f = \int_E f$.

In a measurable partition of an interval, the sum of the measures of the subsets in the partition equals the length of the interval

Sets of null (Lebesgue)-measure and sigma compacts

Every subset of a subspace of $\mathbb{R}^n$ of dim $<n$ has measure 0

$f:[a,b] \rightarrow \Bbb R$ measurable function, $f\geq0$ a.e. If $f^{-1}((0,\infty))$ has measure $>0$ , $f^{-1}((1/n,\infty))$ does for some n.

Hardy Littlewood maximal function $|h(x)|<|h^*(x)|$ for almost every x

Caratheodory measure problem

Countable $\varepsilon$-net of measurable sets

Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra

A "nice result" in Functional Analysis

Bogachev change of variables theorem.

$\mu$, a Borel measure on $\mathbb{R}$ satisfies $\mu(K)<\infty$ for compact K, $\mu(x+E)=\mu(E)$. Then $\mu(E) =c\cdot\lambda(E)$

Prove that $\lim_{n\rightarrow \infty} \int_{[-n,n]} f\,d\lambda= \int f\,d\lambda.$

What is a tube in $\mathbb{R}^n$?

Lebesgue integration: integral of continuous function tends to infinity

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