# Questions tagged [borel-sets]

For questions about Borel sets. Please, add also other tags indicating the area, e.g., (measure-theory), (general-topology), (descriptive-set-theory), etc.

808 questions
Filter by
Sorted by
Tagged with
40 views

### An example of an $L_p$ space with Borel measure and dimension larger than $\mathfrak c$

Given a normed space $X$, we can define the Borel $\sigma$-algebra in $X$ as the smaller $\sigma$-algebra containing all the open (or closed) sets of $X$. Suppose a measure $\mu$ is chosen in the ...
• 136
226 views

### Why is intersection of open sets not closed under countable unions?

I’m learning about Borel sets and I’m struggling to understand the following sentence. The intersection of every sequence of open subsets of $\mathbb{R}$ is a Borel set. However, the set of all such ...
• 595
23 views

### Limit of the measure of the delta neighborhood of Borel Sets.

Statement 1.7 from Fractal Geometry by Falconer I am reading through Fractal Geometry Mathematical Foundations and Analysis by Falconer and I am not very familiar with measure theory. While reading ...
42 views

### Complement of a set of measure zero is dense in $[a,b]$?

Assume we have a subset of measure zero $N$ in the Borel-$\sigma$-field $\mathcal{B}([a,b])$. I think that $[a,b]\setminus N$ is a dense set. But how to prove that? My attempt: let $x \in [a,b]$ and ...
• 219
54 views

### Is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ complete?

Why is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ said to be incomplete wrt. to the Lebesgue measure? As I understand a complete measure space contains all subsets of all sets of measure ...
• 219
1 vote
22 views

103 views

### What is meant by " analytic set "?

I am a beginner with set theory. What is meant by " analytic set " ? I see this term in the context of set theory, calculus and real analysis but I have no idea what it means. See here : ...
• 16k
36 views

48 views

• 125
85 views

### The existence of measurable subsets in non-mesurable sets

Let $\mu$ be a Borel probability measure on $\mathbb{R}$ and let $S\subseteq\mathbb{R}$ be a Borel set such that $\mu\left(S\right)>0$. Is it true that for any $A_1,A_2\subset\mathbb{R}$ that are ...
30 views

### How to prove that $\{(s, t) \in \mathbb R^2 : s \le t \le s+1 \text{ and } t \in A\}$ is a Borel set?

I'm doing an exercise in which I need to apply Fubini–Tonelli theorem on a non-negative function $f$. So I have to check that $f$ is measurable. However, it turns out that applying the definition of ...
• 17.4k
79 views

### Show that if $X,Y$ are topological spaces and $f:X\to Y$ is a continous function , then $f$ is a Borel measurable.

Show that if $X,Y$ are topological spaces and $f:X\to Y$ is a continous function , then $f$ is a Borel measurable. Any help what am I supposed to prove here ? My attempt: $\mathcal{B}(X)$ is the open ...
• 2,322
117 views

### Exercise 1.7.15 from Salamon's Functional Analysis

This problem has three questions (a), (b) and (c). I've done most of them, with a little conclusion in (c) undone, which I've thought about it for a long time. (c) Let $f: X\to Y$ be a Borel ...
• 43
54 views

### Countable generation of the Borel measure algebra

Consider the Borel sigma-algebra on $\mathbb{R}$ quotiented by the ideal of measure-zero sets (see definitions below). This forms a measure algebra. My question is whether this measure algebra is ...
1 vote
31 views

### Whether the function $(x,x')\mapsto\rho\big(f(x),f(x')\big)$ is Borel for a Borel map $f\colon (X,d)\to (Y,\rho)$ with $(Y,\rho)$ not being separable?

Let $(X,d)$ be a compact metric space, let $(Y,\rho)$ be an arbitrary metric space. Let $\mathcal{B}(\mathbb{R}),\mathcal{B}(X),\mathcal{B}(Y),\mathcal{B}(X\times X), \mathcal{B}(Y\times Y)$ denote ...
• 561
90 views

### Intersection of a Borel set and its translation

My idea is to use regularity somehow and use compact sets where this statement clearly holds. This doesn't seem to be working. Is this the right approach? Any help is appreciated.
Let $T:(Z,\|\cdot\|_{Z})\to(H,\|\cdot\|_{H})$ be a continuous linear operator between a separable Banach space, $Z$, and a separable Hilbert space, $H$. Assume that both $Z$ and $H$ are infinite-...
Given $p\in (0,1)$ we define a Borel probability $\mu_p$ in the interval $[0,1]$. We assign $\mu_p([0,1/2))=1-p$ and $\mu_p([1/2,1))=p$. We iterate this process by diving each interval in two and ...