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See also: When is the infimum of an arbitrary family of measurable functions also measurable? My answer is for supremum, but the same holds for infimum since the corresponding results can be obtained …

The relation $$ x\sim y \quad \iff \quad x-y\in\mathbb{Q} $$ is an equivalence relation that partitions $[0,1]$ into countable sets of the form $[t]=(t+\mathbb{Q})\cap [0,1]$. The set of all equivalen …

The answer to Question 1 is yes. This is a result of Falconer, Corollary 6.6 in K. J. Falconer, Sets with prescribed projections and Nikodým sets. Proc. London Math. Soc. (3) 53 (1986), no. 1, 48–64 …

There are strictly increasing $C^1$ functions that map sets of positive measure to sets of measure zero. Here is a construction: Let $C\subset [0,1]$ be a Cantor set of positive measure. For a constr …

The answer is yes. I wrote a proof using YCor's comment. Theorem. There a finitely additive measure defined on all subsets of positive integers $\mathbb{N}$, with values into $\{0,1\}$, (only two …

It is false without assuming that the measure is $\sigma$-finite. Let $\mu$ be the counting measure on $\mathbb{R}$. Then $h>0$ a.e. means $h>0$ everywhere and clearly $$ \int_{\mathbb{R}}hd\mu=\sum_{ …

Take a set $K\subset\mathbb{R}^2$ that is homeomorphic to the Cantor set and has positive $2$-dimensional Lebesgue measure. Take a homeomorphism of the ternary Canor set onto $K$ and extend it as a pi …

I have an example which does not strictly answer the question, but a slightly weaker one. Theorem 1. There is a function $f:\mathbb{R}^n\to\mathbb{R}$ that is approximately Fréchet differentiable alm …

The function $f(x)$ need not be decreasing or even continuous. Let $S$ be the union of three segments in $\mathbb{R}^2$. $$ S=([0,0.1]\times\{ 0\})\cup(\{0\}\times[0,1])\cup (\{0.1\}\times[0,1]). $$ …

A measurable function $f:[0,1]\to\mathbb{R}$ maps Lebesgue measurable sets to measurable sets if and only if it has a Lusin property: the image of a set of measure zero has measure zero. Here is an …

By the Dirichlet principle harmonic functions in $W^{1,2}(B_4)$ minimize the Dirichlet energy $\int_{B_4}|\nabla u|^2$ among all function in $W^{1,2}(B_4)$ with the same boundary data. Since $v\in W^{ …

This is not a full answer, but some comments that might put you on the right track. If a function $f=(f_1,\ldots,f_n):\mathbb{R}^n\to\mathbb{R}^n$ is sufficiently smooth and has compact support, the …

The infimum of any family of measurable functions is measurable if we interpret the infimum as the lattice infimum. For details see: https://mathoverflow.net/a/316658/121665

This is a partial answer. In $\mathbb{R}^n$, $\mathcal{H}^n$ coincides with the Lebesgue measure so $\mu(\mathcal{H}^n)$ coincides with the $\sigma$-algebra of Lebesgue measurable sets.

I will mention seven different applications: Characterization of almost everywhere differentiability. The following result is a consequence of the Rademacher theorem: Theorem (Stepanov). A functi …