# Questions tagged [geometric-measure-theory]

The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential geometry, Riemannian geomerty, sub-Riemannian geometry, as well as calculus of variations. Statements such as the isoperimetric inequality and the coarea formula, and subjects such as the Plateau problem belong under this tag.

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### A consequence of Huisken's rescaled monotonicity formula : Stone's Lemma

The context is that of the mean curvature flow, more precisely, concerning Type I singularities and the rescaling procedure. The text I am following is by Mantegazza: "Lecture notes on Mean ...
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### Reference for currents in geometric measure theory - continuous version?

Let $M$ be a Riemannian manifold. I am interested in the dual of the space of continuous (not necessarily smooth) vector field on $M$. The dual of the space of smooth vector field would be the space ...
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### Proof that a Lipschitz function is a.e. approximately differentiable by using it is "almost" $C^1$

I am trying to do the following exercise: to prove that a Lipshitz function $f: \mathbb{R}^n \mapsto \mathbb{R}^m$ is $\mathcal{L}^n$-almost everywhere approximately differentiable using the following ...
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### Proving Lebesgue dominated convergence theorem using Egorov theorem

I'm trying to prove Dominated Convergence Theorem (DCT) by using Egorov's theorem. And so far, I know how to do it when the functions are defined in finite measure set in real. Is it possible to also ...
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### Hausdorff measure and uniform subdivisions

Sorry I never got to learn about Hausdorff measures so this may come as simple. Suppose $$A = \{x\in [0,1)^d: f(x)=0\}$$ where $f$ is analytic and nontrivial. Then $A$ is an analytic variety of ...
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### Is the Lebesgue measure of the boundary of a union of $n$-balls zero?

I'm working in $\mathbb{R}^n$, but I believe that the question is relevant in any metric space. Let $B_1^n$ be the closed ball of radius 1 centered at the origin, and let $\oplus$ denote Minkowski ...
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### If there is a zero measure set with positive measure pre-image, is there a level set with positive measure?

Let $f:\mathbb{R}\to\mathbb{R}$ be smooth (I suppose $C^1$ should be enough, but I would be happy with an answer when $f\in C^\infty$). Assume that there exists a Borel set $A\subset\mathbb{R}$ with ...
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We have the following theorem: Let $f: A \subset \mathbb{R}^n \to \mathbb{R}^m$ be Lipschitz, with Lipschitz constant Lip $f$, $1 \leq n \leq m$. Then we have $$\int_{\mathbb{R}^m} H^0(f^{-1}(y) \cap ... 1 vote 0 answers 17 views ### Applying Hausdorff Area formula with multiplicity I am trying to understand the following formula. Let f: A \subset \mathbb{R}^n \to \mathbb{R}^m be Lipschitz, with Lipschitz constant Lip f, 1 \leq n \leq m. Then we have$$ \int_{\mathbb{R}^m} ...
Given a measurable function $f : \mathbb{R}^n \to \mathbb{C}$, we can define the Hardy-Littlewood maximal operator $M$ by Mf (x) = \sup_{0 < r < \infty} \mu(B(x, r))^{-1} \int_{B(x, r)} | f | \...