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Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
13
votes
3
answers
816
views
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose pr...
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The projecti …
10
votes
2
answers
1k
views
Can the integration of integrable sections of a measurable function of two variables ever re...
I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct …
3
votes
1
answer
144
views
Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact H...
The following problem is a stumbling block in a research project that I am working on:
Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it tru …
3
votes
2
answers
457
views
If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $...
This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer provi …
5
votes
1
answer
291
views
If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bull...
Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise limi …
6
votes
0
answers
361
views
Approximating a measurable function from a second-countable, locally compact Hausdorff group...
Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere po …
4
votes
1
answer
794
views
The Notion of Strong Measurability for Separable Banach Spaces
Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-e …