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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 …
Transcendental's user avatar
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 …
Transcendental's user avatar
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 …
Transcendental's user avatar
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 …
Transcendental's user avatar
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 …
Transcendental's user avatar
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 …
Transcendental's user avatar
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 …
Transcendental's user avatar