# Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

2,907 questions
Filter by
Sorted by
Tagged with
0 votes
0 answers
55 views

4 votes
2 answers
370 views

### A possible measure-theoretic pathology

Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$ that has the following "shadow property": For any aligned open square $C = A \times B$ that intersects $S$, ...
• 41
0 votes
0 answers
95 views

### Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$

I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion. I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
• 11
1 vote
0 answers
63 views

### Benjamini-Schramm convergence: convergence on metric balls implies weak convergence?

In the famous paper by Benjamini-Schramm 2001, they consider the space of rooted graphs with uniformly bounded degrees, this space modulo rooted graph isomorphism is denoted by $\mathcal X$. This ...
• 492
-1 votes
1 answer
298 views

### The category Prob of finite measure spaces does not admit all products [closed]

I am currently working in a category called Prob which has objects which are finite measure spaces and morphisms which are measure preserving maps between the spaces. A map $f:X\to Y$ is measure ...
• 91
5 votes
1 answer
381 views

4 votes
0 answers
162 views

### Finding balls with big measure

Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$. $$\mu(B(x,r)) < \delta r^n.$$ Under ...
2 votes
0 answers
46 views

• 233