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Results tagged with measure-theory
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user 32507
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
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 …
- 1,724
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.
- 1,724
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,
…
- 1,724
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 …
- 1,724
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^ …
- 1,724