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Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

1 vote

Envelopes of functions with respect to some convex cone $\mathcal{F}$

If you switch from your perspective of functions to their epigraph, I think that you end up with a "closure operator", see Wikipedia. I am not sure about other examples, but if $\mathcal{F}$ consists …
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1 vote
Accepted

Sequential compactness of a sequence of curves of Borel probability measures

Here are some ideas, but I did not have the time to check every detail. Let $T \subset [0,1]$ be countable and dense. Using a diagonal sequence argument, one can find a subsequence such that $$ \mu_n^ …
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2 votes

Decomposition of non negative Radon measure into $L^1$ and $H^{-1}$ functions

You can find both results in Theorems 2.1 and 2.4 in Lucio Boccardo, Thierry Gallouët, Luigi Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, …
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5 votes

Measurability of essential supremum of function of two variables

For the record, here is a simple answer based on Tonelli/Fubini's theorem. For arbitrary $c \in \mathbb R$, the set $M_c := \{(x,y) \in X \times X \mid f(x,y) > c\}$ is measurable, thus the indicator …
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3 votes
Accepted

sub and super-levelset regularity for Sobolev functions

One positive answer is that this set is $p$-quasi-open, see some resource about capacity theory, e.g., here: https://math.stackexchange.com/questions/48776/capacity-theory-beginner-resources.
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