Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
20 questions from the last 30 days
16
votes
2
answers
989
views
Defining Lebesgue non-measurable sets with countable information
Is there a formula $\phi$ in the language of set theory such that
$$
\text{ZFC proves } \exists x \in \mathbb{R}:\text{ the set }A_x:=\{y\in\mathbb{R}:\phi(x,y)\} \text{ is not Lebesgue measurable?}
$...
- 237
11
votes
2
answers
1k
views
Which sets are "persistently measurable"
Let $\mathcal{E}$ be the class of Lebesgue-measurable subsets of $\mathbb{R}$. The notion of $(\mathcal{E},\mathcal{E})$-measurable function is a bit pathological since lots of continuous functions ...
- 19.4k
4
votes
1
answer
270
views
Nonseparable Hoffmann-Jørgensen metric space
A metric space $(X,d)$ satisfies the Hoffmann-Jørgensen (HJ) property if for any two Borel measures $\mu_1,\mu_2$ we have that $\mu_1(B_r(x))=\mu_2(B_r(x))$ for all $r>0$ and $x\in X$ implies $\...
- 2,345
5
votes
1
answer
177
views
Integral representation of Markov operators
On a measurable space $(E,\mathcal E)$, a stochastic kernel is a function $p\colon E\times \mathcal E\to [0,1]$ such that:
for each $x\in E$, the function $A\mapsto p(x,A)$ is a probability measure;
...
- 286
4
votes
1
answer
252
views
Does anyone use measures that take values in real numbers and cardinal numbers?
The counting measure on $\mathbb{R}^n$ is a map that takes a subset $A$ of $\mathbb{R}^n$ and returns its cardinality if it is finite or the symbol $\infty$ if it is infinite.
So, if $A\subseteq\...
- 1,038
4
votes
1
answer
201
views
Minimal dominating measure for dominated Markov kernel
Let $(X,\mathcal X)$ and $(Y,\mathcal Y)$ be measurable spaces, $\pi \colon \mathcal Y\times X \to [0,1]$ a Markov kernel. We assume that it is measurably dominated, i.e. there is a $\sigma$-finite ...
- 105
1
vote
1
answer
139
views
A question on the Banach space property of a rearrangement invariant function space
Consider a measure space $(S,\mu)$ and assume that $\mu(S)=1$. We consider the quantile function (or nonincreasing rearrangement) of a real valued function $f:S\to\mathbb{R}$ as the function
\begin{...
- 169
3
votes
0
answers
251
views
Consistency of a measure witnessing a strengthening of Freiling’s axiom of symmetry
I'm interested in Freiling's axiom of symmetry and I specifically wonder if it may be proven from more basic axioms about measures on $\mathbb R^n$, in the sense that there is a sequence of measures $\...
- 41
1
vote
1
answer
154
views
Let $k \mapsto f_k$ have nonnegative derivative in $L^0(\mu)$, then it is increasing almost everywhere
Let $\mu$ be a finite measure on some measurable space $(X, \Sigma)$ and consider the topological vector space $L^0(\mu)$ of all real-valued measurable functions on $X$ with respect to convergence in ...
- 713
1
vote
1
answer
167
views
Measurability of a subset of the plane with respect to the product of Lebesgue sigma-algebras
Let $\mathfrak{L}$ denote the $\sigma$-algebra of Lebesgue-measurable subsets of $\mathbb{R}$. Consider a null subset $\mathcal{N} \in \mathfrak{L}$. Does the subset of $\mathbb{R}^{2}$
$$
\bigcup_{t \...
- 912
11
votes
0
answers
176
views
Reverse mathematics of $\mathbb{\Sigma^1_2}$-measurability
Analytic sets are projections of Borel sets, and are known to be Lebesgue measurable (in fact universally measurable). The question of whether measurability of analytic sets can be shown in some ...
- 651
2
votes
1
answer
125
views
How to prove the convergence of the maximum point random variable of random concave function sequence？
I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
- 35
2
votes
1
answer
133
views
Is there a increasing, convex, superlinear $f$ with $c_1 f(x)y \leq f(xy)\leq c_2 f(x)f(y)$ such that $\mathbb{E}[f(X)] < \infty$?
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random ...
- 784
2
votes
1
answer
118
views
When is a mapping that is both a measure isomorphism mod 0 and an order isomorphism unique mod 0?
Suppose we have two measure spaces, $(\Omega, \mathscr{F}, \mu)$ and $(\Psi, \mathscr{G}, \nu)$, with $\mu(\Omega) = \nu(\Psi) < \infty$, and we consider the set of measure isomorphisms mod 0 ...
- 319
6
votes
0
answers
144
views
Ergodicity in the Wiener-Wintner Ergodic Theorem [cross-post from MSE]
I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
- 61