# Tagged Questions

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### A Borel measure defines semi-continuous function?

Let $X$ be a metric space with outer measure $\mu$, which is assumed to be a Borel measure, i.e., all Borel sets are measurable. For a fixed subset $A\subset X$ (not necessarily measurable, but you ...
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### Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
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### Why Steiner Symmetrization makes a measurable set to a measurable one?

I find the Steiner Symmetrization is very useful in proving that the Hausdorff measure coincide with Lebesgue in the Euclidean space. However, I never saw anybody mention that the Steiner ...
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### What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
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### BV Function times characterstic function still BV?

I am wondering: given a function $u \in BV(\Theta)$ where $\Theta$ is an open subset in $\mathbb{R}^n$ and a Borel subset $B \in \mathcal{B(\Theta)},$ is the function $w \colon= u \chi_B$ still in ...
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### Aproximating a function on SO(3)

By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from ...
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### Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: Given $k<n$ every $k$-dimensional subspace of $\mathbb{R}^{n}$ has ...
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### $L^1$ approximate continuity points of epigraph of a continuous function

Consider a continuous function $f\colon \mathbb R\to \mathbb R$. Let $F(x,y):=\mathrm{sign\,}(y-f(x))$. Is it true that for a.e. $x\in \mathbb R$ the point $(x,f(x))$ is not a point of $L^1$ ...
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### Prove that $\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$

Suppose that the measurable sets $A_1,A_2,...$ are "almost disjoint" in the sense that $\mu(A_i\cap A_j) = 0$ if $i\neq j$. Prove that $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ ...
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### change of variable for Hausdorff measure

Let $U$ be an open set in $\mathbb{R}^n$, $T:U \rightarrow \mathbb{R}^n$ be a diffeomorphism defined on $U$, $\mu$ be the $d$-dimensional Hausdorff measure, $0<d<n$. Is it true that we have the ...
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### Alternative rigorous definition of a surface integral

Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
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### About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
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### Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
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### Approximate tangent planes and densities

I'm studying the proof of the following theorem (Simon 'Lectures on Geometric measure theory' Theorem 11.6): Theorem Let $M \subset R^{n+k}$ be $H^n$-measurable (where $H^n$ is the Haussdorf ...
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### Homogenous measure on the positive real halfline

Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0}$, we have $\mu(aE)= \mu(E)$, where ...
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### Radon-Nikodým derivative with respect to the Lebesgue/Hausdorff measure is always defined

Question 1. Is it possible to say that the Radon-NikodĂ˝m derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized ...
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I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\ The proof can be summarized in the following steps:\ 1)For every ...
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### A question about Lebesgue measure 3

Let $L^k$ be the k-dimensional lebesgue measure. Let $A \subset R^n$ be a Borel set. Suppose we have proved that $L^1(A \cap l )=0$ for each line $l$ parallel to some line passing throught the ...
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### Measurability of a certain set in Falcolner's Geometry of Fractal Sets

On page 24 of Falcolner's The Geometry of Fractal Sets, Falcolner defines the set $F = \{ x \in E : \mathcal{H}^s(E \cap U) < \alpha$ diam$(U)^s$, for all convex sets $U$ containing $x$ such that ...
I am working through Evan's book on Weak Convergence Methods for Nonlinear PDE. He assumes that $U$ is an open bounded smooth subset of $\mathbb{R}^n$ and that $1<q<n$. In ($\S$D) concerning ...