# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

23 views

119 views
+500

### Limit of uniformly converging volume-preserving homeomorphisms

Definition A continuous map $f\colon \mathbb{R}^n \to \mathbb{R}^n$ is volume-preserving if, for every Borel set $V\subset\mathbb{R}^n$, $\mathcal{L}^n(V) = \mathcal{L}^n(f^{-1}(V))$. I am wondering ...
23 views

### Every metric $\delta$ on sets is $\delta (A, B) = \mu(A \Delta B)$, resp. $\delta(A, B) = \mu(A \Delta B)/\mu (A \cup B)$, for some measure $\mu$?

Suppose $\mu$ is a measure and that $\delta (A, B) = \mu(A \Delta B)$ (where $\Delta$ represents symmetric difference, and $A=B$ whenever $\mu(A \Delta B) = 0$). Then $\delta$ is a distance function. ...
26 views

20 views

### What's $\{g(\theta^n x)\}$ sequence called?

Let $(S, A, µ)$ be a probability space and $g$ be a measurable function on it. Let $\theta$ be a µ-measure preserving transformation on it. If $\theta$ is a ergodic, what's $\{g(\theta^n x)\}$ ...
33 views

5k views

### Bounded Function Which is Not Riemann Integrable

This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak. Give an example of a bounded function ...
Let $\{f_{n}\}$ be a sequence of functions in $L^1(\mathbb{R})$ such that $\displaystyle \sum_{n=1}^\infty\|f\|_{1}<\infty.$ Show that $$f(x): = \sum_{n=1}^\infty f_n(x)\text{ converges a.e., }\, f\... 0answers 46 views ### Is a measure on product space necessarily a product of measures? Let X,Y be some nice measureable spaces (i'm interested in [0,1] so we can assume compact, etc.). let \mu be a measure on X\times Y.(again, assume it's nice, i.e. probability measure. anything ... 0answers 20 views ### Proving that A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\} is an inner measure Let (X,\Sigma, \mu) be a measure space and define m: 2^X \to [0,\infty] by m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}. Show that m is an inner measure. There are 4 ... 1answer 29 views ### Lebesgue decomposition and Radon-Nikodym derivative given a function. Define f:\mathbb{R}\rightarrow\mathbb{R} by$$f(x)=\begin{cases}0 & \text{ if }-\infty<x<0\\ 1 &\text{ if }0\leq x <1\\ x^{2}+x^{3} &\text{ if }1\leq x <2\\ 17 &\text{ if ...
From Wikipedia Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space \$(...