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I will mention seven different applications: Characterization of almost everywhere differentiability. The following result is a consequence of the Rademacher theorem: Theorem (Stepanov). A function ...
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Are functions of bounded variation a.e. differentiable?

No. Take a dense countable set $\{x_1,x_2,\dots\}$ in $\mathbb{R}^d$ and a sequence $(r_i)\subseteq\mathbb{R}^+$ such that $\sum_i r_i^{d-1}<\infty$. Then the function f=1_{\bigcup_{i=1}^\infty ...
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Why do almost all points in the unit interval have Kolmogorov complexity 1?

I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...
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Generalized Stokes' theorem

When I wrote that Wikipedia paragraph, I think I had in mind Theorem 5.16 in Evans & Gariepy, Measure Theory and the Fine Properties of Functions, which essentially proves Stokes' theorem for ...
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A gerrymandering problem - can you always turn a tie into a landslide victory?

Yes, the almost partition exists. Instead of letting $\mu(E)\geq\frac{\mu(\Omega)}{2}$, I let $\mu(E)\in(0,\mu(\Omega))$ be arbitrary and proved that you can divide $\Omega$ into $N$ open simply ...
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Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?

This is true, $f$ is actually everywhere differentiable. It is the "Limit under the Sign of Derivative" Theorem; it also holds for sequences of maps between Banach spaces (and you may even ...
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How badly can the Lebesgue differentiation theorem fail?

Metafune has given an example of the limit failing to be $0$ at a particular point - namely for $n > 1$, the function $|x|^{-\alpha}$, with $1 \leq \alpha < n$ has that limit equal to $\infty$ ...
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Almgren's mimeographed lectures notes on varifolds

I just bumped into your post after doing some google search to find a bibtex entry for Almgren's notes. I have a copy of them: I could copy it and send it to you (or maybe scan it and share it via ...
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Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

That is true. Caccioppoli sets are also known as sets of finite perimeter. Theorem. Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the ...
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Existence of subset with given Hausdorff dimension

First of all, $\dim_{H} (A) = \alpha$ iff $H^k(A)=\infty$ for all $k<\beta$ and $H^k(A) = 0$ for all $k>\beta$. Then $H^\alpha(A) = \infty$ for all $\alpha \in (0,\beta)$. If $A$ is closed ...
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Generalized Stokes' theorem

Harrison, Jenny, Stokes' theorem for nonsmooth chains. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 235â€“242. This research announcement reports progress in developing a viable theory of integration ...
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Proper homotopy

The following counterexample is stolen from page 1 of this paper by Thomas Rot. The map $[0,1]\times\mathbb R\to\mathbb R$ given by $(t,x)\to (1-t)x^2+x$ is not proper, e.g., the preimage of $\{0\}$ ...
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Metric measure spaces: in what sense is analysis on these spaces "non-smooth"

I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available here. The beginning of the introduction reads: "The word nonsmooth in the title refers both to functions and ...
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A Besicovitch-type Covering Theorem

This trivial counterexample in $\mathbb R^2$ should have taken me five minutes. Instead, I spent almost two days. The moral is the usual one: after 50 you'd better give up on mathematics. Let $y,z$ ...
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Is the composition of two nowhere differentiable functions still nowhere differentiable?

The composition may have points of differentiability. Let $f_0(x)=x$ for $x\geq 0$ and $f_0(x)=2x$ for $x<0$. Let $g_0(x)=2x$ for $x\geq 0$ and $g_0(x)=x$ for $x<0$. Then none of them is ...
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Tiling the plane with finitely many congruent pieces

Write $A_i=T_i(A)$ for $A=A_1$, where each $T_i$ is a rigid motion. For each $i$, we have $|A_i\cap B(r)|=|T_i(A\cap T_i^{-1}(B(r))|=|A\cap T_i^{-1}(B(r))|$. The symmetric difference between this set ...
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If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected

Yes, $\mathbb{R}^n\setminus E$ has to be path-connected. Let $x,y\in\mathbb{R}^n\setminus E$, we will prove that there are many paths going from $x$ to $y$ inside $\mathbb{R}^n\setminus E$. We can ...
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Uncountable collections of distinct subsets of an interval (existence)

My comment reposted as an answer: If the continuum hypothesis holds, then we can give a well order $\prec$ to $\mathbb{R}$ isomorphic to the first uncountable ordinal. And then for each $j\in[-1,1]$ ...
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Fractals of dimension zero

One example: the set of Liouville numbers has Hausdorff dimension zero. In number theory, a Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, ...
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Ergodicity and mixing of geodesic and horocyclic flows

The algebraic approach is just one of many ways to deal with the geodesic and horocycle flows. Also your quick summary does not really pay tribute to the many ways to deal with it using "...
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Derivative of distance function to a closed, rectifiable set

The distance $f: x \mapsto \mathsf{dist}(x,\Gamma)$ to a closed set $\Gamma$ in $\mathbb{R}^n$ is differentiable in $x \notin \Gamma$ iff the nearest point projection is unique; denote this by \$x_\...
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