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Results tagged with measure-theory
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user 1004
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
29
votes
Is there a measure zero set which isn't meagre?
There's already been some good answers to this. However, this is something that I have also thought about recently, because I happen to have come across several meagre sets of full Lebesgue measure in …
- 17.1k
26
votes
Non-Borel sets without axiom of choice
Measure theory without the Axiom of Choice (not even countable choice) is discussed in Fremlin, Measure Theory, Volume 5, Chapter 56. This is freely available online. Thanks to MO and ex-falso-quodlib …
- 17.1k
23
votes
Accepted
Is arbitrary union of closed balls in $\mathbb{R}^n$ Lebesgue measurable?
No, in dimension $N>1$, it does not have to be Borel measurable. E.g., in 2 dimensions, consider, a non Borel measurable subset of the reals $S$, and let $A$ be the union of closed unit balls centered …
- 17.1k
17
votes
Accepted
Bochner integral of stochastic process = path by path Lebesgue integral?
Yes, the Bochner integral does agree with the Lebesgue integral of the sample paths of the process. We can prove this in a slightly more general situation than that asked for in the question.
For a p …
- 17.1k
13
votes
Accepted
What is a Gaussian measure?
You could alternatively try defining Gaussian measures as $2$-stable distributions. This does remove any reliance on finite dimensional projections, and even removes reference to topology. Let $V$ be …
- 17.1k
10
votes
Does there exist an event independent of a given sigma-algebra?
No. If the probability space contains atoms then you can easily construct sub-$\sigma$-algebras which are not independent to any nontrivial events.
However, we can still construct examples, even for …
- 17.1k
9
votes
Accepted
Is every probability space a factor space of the Haar Measure on some group?
It is possible to find the following: A compact abelian group G with Haar measure $\mu_G$, a subset $S\subseteq G$ of full outer Haar measure and a measurable function $f\colon S\to X$ with $\mu_P(E)= …
- 17.1k
5
votes
Accepted
Non-existence of integral with respect to Poisson Random Measure
As I mentioned in my comment, you can prove the statement and its converse by looking at the moment generating function. Supposing that f ≥ 0 and λ > 0 is a real number, the following is true for a Po …
- 17.1k
4
votes
Borel kernel over an analytic set implies existence of a Borel map
The following is a construction of an analytic set $A\subseteq\omega^\omega\times\omega^\omega$ whose vertical sections exclude at most a single point, and which does not admit a Borel uniformization. …
- 17.1k
3
votes
Accepted
Existence of dominating measure for weak*-compact set of measures
There always exists a dominating measure.
First, given two finite measures $\mu,\nu$ on $(\Omega,\mathcal{F})$, the Lebesgue decomposition theorem says that there is an $A\in\mathcal{F}$ such that $1 …
- 17.1k
1
vote
Completeness of Borel measure
No, it is not possible for $\mu$ to be complete.
There exists a closed subset $K$ of $X$ with $\mu(K)=0$ and a continuous onto map $f\colon K\to2^\omega$.
With $K,f$ as above, if $A\subseteq …
- 17.1k