Question

How to prove that $\int_0^1 dx \int_0^x f(x,y) dy = \int_0^1 dy \int_y^1 f(x,y) dx$ for a continuous function $f$ the given interval

score 117 · Answer 1 · 2018-02-07 07:34:20

117

Views

How to prove that $\int_0^1 dx \int_0^x f(x,y) dy = \int_0^1 dy \int_y^1 f(x,y) dx$ for a continuous function $f$ the given interval

Published on 07 Feb 2018 - 7:34

score 131 · Answer 2 · 2026-03-19 22:58:13

131

Views

Is it true $\mu (A+B) = \mu (A) + \mu(B) $ for bounded sets $ A, B \subset R $

Published on 19 Mar 2026 - 22:58

score 200 · Answer 3 · 2018-02-07 23:17:54

200

Views

If $|E|=0$ then $f$ is measurable

Published on 07 Feb 2018 - 23:17

score 679 · Answer 4 · 2026-03-22 18:24:18

679

Views

For $E\subset\mathbb R^n$ with Lebesgue outer measure $0<m^(E)<\infty$, $\alpha\in(0,1)$, there is a ball $B$ with $m^(E\cap B)\ge \alpha m^*(B)$.

Published on 22 Mar 2026 - 18:24

score 53 · Answer 5 · 2026-03-20 06:16:57

53

Views

Lebesgue's monotone convergence theorem, - boundedness

Published on 20 Mar 2026 - 6:16

score 236 · Answer 6 · 2026-03-23 02:49:38

236

Views

Consider the operator $T:L^2(0,1)\rightarrow L^2(0,1)$. Is it well defined, linear, bounded, compact?

Published on 23 Mar 2026 - 2:49

score 1k · Answer 7 · 2026-03-22 19:37:40

1k

Views

Radon-Nikodym derivative of discrete measure

Published on 22 Mar 2026 - 19:37

score 219 · Answer 8 · 2018-02-12 05:32:04

219

Views

Borel set with constant positive but not full measure in each interval.

Published on 12 Feb 2018 - 5:32

score 32 · Answer 9 · 2026-03-17 20:56:21

32

Views

Prove that for any $k \in \Bbb{R}$ we have$ \quad \mu^{}(kA) = |k|\mu^{}A, $

Published on 17 Mar 2026 - 20:56

score 510 · Answer 10 · 2018-02-12 23:04:24

510

Views

Borel set with translation invariant measure in each interval of any length

Published on 12 Feb 2018 - 23:04

score 64 · Answer 11 · 2018-02-13 13:36:11

64

Views

A measure space which is not finite but implies $L^p$ inclusion

Published on 13 Feb 2018 - 13:36

score 56 · Answer 12 · 2018-02-14 08:44:30

56

Views

Sum of a zero-measure sets

Published on 14 Feb 2018 - 8:44

score 56 · Answer 13 · 2018-02-14 15:04:15

56

Views

Need for a set to have finite Lebesgue measure.

Published on 14 Feb 2018 - 15:04

score 378 · Answer 14 · 2018-02-15 07:22:06

378

Views

Length is a Pre-Measure on the Semi-Ring of Bounded Half-Open Real Intervals

Published on 15 Feb 2018 - 7:22

score 91 · Answer 15 · 2018-02-15 23:14:01

91

Views

Do all measure zero sets belong to the measure space: If $\mu(N_1) = 0$ then is $N_1 \in \mathcal{M}_{\mu}$?

Published on 15 Feb 2018 - 23:14

List Question

How to prove that $\int_0^1 dx \int_0^x f(x,y) dy = \int_0^1 dy \int_y^1 f(x,y) dx$ for a continuous function $f$ the given interval

Is it true $\mu (A+B) = \mu (A) + \mu(B) $ for bounded sets $ A, B \subset R $

If $|E|=0$ then $f$ is measurable

For $E\subset\mathbb R^n$ with Lebesgue outer measure $0<m^(E)<\infty$, $\alpha\in(0,1)$, there is a ball $B$ with $m^(E\cap B)\ge \alpha m^*(B)$.

Lebesgue's monotone convergence theorem, - boundedness

Consider the operator $T:L^2(0,1)\rightarrow L^2(0,1)$. Is it well defined, linear, bounded, compact?

Radon-Nikodym derivative of discrete measure

Borel set with constant positive but not full measure in each interval.

Prove that for any $k \in \Bbb{R}$ we have$ \quad \mu^{}(kA) = |k|\mu^{}A, $

Borel set with translation invariant measure in each interval of any length

A measure space which is not finite but implies $L^p$ inclusion

Sum of a zero-measure sets

Need for a set to have finite Lebesgue measure.

Length is a Pre-Measure on the Semi-Ring of Bounded Half-Open Real Intervals

Do all measure zero sets belong to the measure space: If $\mu(N_1) = 0$ then is $N_1 \in \mathcal{M}_{\mu}$?

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