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Results tagged with measure-theory
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user 51695
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
8
votes
Accepted
Riemann rearrangement theorem for $L^1$ functions
The problem is trickier than I initially thought, but with the corrected condition it can be done. I need to assume that $\sum c_n \delta_n$ is conditionally but not absolutely convergent, $0 < \delta …
- 2,504
6
votes
What is Young measure?
Related to Nate River's answer, I personally prefer to think of Young measures as single measures $\nu$ on $U \times \mathbb{R}^m$, which have the condition that their projection on the first componen …
- 2,504
4
votes
Accepted
Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
In Evans & Gariepy's "Measure theory and fine properties of functions", Sec. 5.3., they construct the trace operator on a bounded Lipschitz domain $\Omega$ for BV-functions (and thus by inclusion for …
- 2,504
4
votes
Average of the sum of dirac measures
I assume that by maximal you mean with respect to inclusion. Then the answer is no. Consider the following counterexample on the real line:
Let $\mathcal{B}_\epsilon := \epsilon\mathbb{Z}$ and $\widet …
- 2,504
2
votes
How to interpret this quote of Lin?
This is not a full answer, since I do not know the counterexample Lin refers to, but I can offer some explanations and guesses which are too long for a comment:
You can define a first variation for cu …
- 2,504
1
vote
Equivalence of statements about level sets: $u|_{S \times [\tau, \infty)}$ depends only on $...
If I am not missing something then the corrected statements look equivalent to me on the level of sets, no need for smoothness.
Fix $\tau > 0$ and assume that 1. holds. For any $t\geq \tau$ and $y \in …
- 2,504
0
votes
Accepted
Distance function and geometry of the set
Since Pietro Majer definitely was right about the structure of the set but hasn't supplied a proof let me jump in with an elementary one. I think the problem is to specific to find a reference, but it …
- 2,504