# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### Change of Variable Proof in Folland

I am reviewing Folland's proof of the following standard result and I have a question on one part. Suppose $\Omega$ is an open set in $\mathbb R^{n}$; $G:\Omega \to \mathbb R^{n}$ is a diffeomorphism ...
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### Another question about proving Lebesgue Decomposition

Note: This is my original question. I have been kindly helped to turn this into a correct proof, which I have posted as an answer so this question won't show up as "unanswered". As an exercise, I am ...
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### Lipschitz transformation maps measure zero sets to measure zero sets. [duplicate]

Let $T:\mathbb{R^2} \to \mathbb{R^2}$ be Lipschitz function. Then, (a) If $E$ is a set in $\mathbb{R}^2$ with Lebesgue measure zero, then $T(E)$ has measure zero in $\mathbb{R}^2$. (B) If $A$...
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### Reference request: product Borel $\sigma$-algebra of non-separable metric spaces

The following is a proposition in Folland's Real Analysis about product sigma algebra: Here $\mathcal{B}_X$ denotes the Borel $\sigma$-algebra on $X$. Could anyone come up with an example that ...
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### An exemple of strict inequality for reverse inequality Minkowski for space $L^p$, $0 < p <1$

Let be $0<p<1$. Suppose that we know that $$\bigg(\int (u + v)^p\bigg)^{1/p} \geq \bigg(\int (u)^p\bigg)^{1/p} +\bigg(\int (v)^p\bigg)^{1/p}$$ for all $u,v \geq 0$ in $L^p$. I need find an ...
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### Does a set of positive measure in a product $\sigma$-algebra contain a rectangular set

Suppose $(E_i, \mathcal E_i)$, $i = 1, \dots, n$, are measurable spaces and let $E := E_1 \times \dots \times E_n$, equipped with the product $\sigma$-algebra, denoted by $\mathcal E$. Suppose $\psi$ ...
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### Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set?

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set? First off, I know that $A$ is a linear map, and a ...
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