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Results tagged with measure-theory
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user 16537
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3
votes
2
answers
647
views
Continuity/measurability of a complicated extension of a family of continuous functions
Bonjour/bonsoir à tous et à toutes.
I've two questions related to something on which I'm working. I've already tried to discuss about them elsewhere, but it hasn't been fruitful so far.
Edit (4 Dic …
- 10.5k
2
votes
0
answers
205
views
On the use of the term "field of sets" in Maharam's papers
I am reading some papers by D. Maharam, and feel a little bit confused about her use of the term "field of sets". Nowadays, I think the term is standardly used to mean a pair $(X, \mathscr{F})$ for wh …
- 10.5k
13
votes
1
answer
1k
views
Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
Question. Is it consistent with ZF that every (countably additive, non-negative) measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on a given set $X$, extends to a (countably addit …
- 10.5k
7
votes
Accepted
A result of Sierpiński on non-atomic measures
I don't yet have a reference, but it seems the result might have been first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:
…
- 10.5k
10
votes
2
answers
2k
views
A result of Sierpiński on non-atomic measures
There is a classical result commonly attributed to W. Sierpiński that reads as follows:
Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ …
- 10.5k
2
votes
Reference for a strong intermediate value theorem for measures
It is also a special case of Theorem 15 (p. 43) in:
A. Fryszkowski, Fixed Point Theory for Decomposable Sets, Topological Fixed Point Theory and Its Applications 2, Dordrecht: Kluwer Academic Publ …
- 10.5k
8
votes
2
answers
560
views
Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?
A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on …
- 10.5k
3
votes
0
answers
234
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the w …
- 10.5k