Questions tagged [borel-sets]

Question 1

I'm reading Sheldon Axler's MIRA book and have a question about understanding the red underlined sentence in this proof: Here is what I think about it: Let $\mathcal{A}$ be the set of finite unions of ...

Question 2

Let $(X, \mathcal{B}(X))$ and $(Y, \mathcal{B}(Y))$ be Polish spaces. Let $f:(X, \mathcal{B}(X))\rightarrow (Y, \mathcal{B}(Y))$ be $\mathcal{B}(X)$-$\mathcal{B}(Y)$-measurable and injective. ...

Question 3

Let $X$ be a Hausdorff topological space and let $\mathcal K$ denote the family of compact subsets of $X$. Assume that the Borel $\sigma$-algebra $\mathcal B(X)$ is generated by compact sets, i.e. $$ \...

Question 4

So I accidentally posted this problem on MO that turned out to be standard: https://mathoverflow.net/questions/501448/example-of-connected-locally-connected-metric-space-that-isnt-path-connected In ...

Question 5

As a corollary of the Sierpiński–Dynkin's $\pi$-$\lambda$ theorem, we have that lemma on the uniqueness of measures defined on a generating $\pi$-system: Lemma (Uniqueness of Measures, $\sigma$-...

Question 6

In the book "Probability and Measure by Robert B. Ash", to construct a measure on the Borel sets of the extended real line it first defines the measure on the right semi closed intervals as ...

Question 7

I tried to solve two problems on my own. I had looked for other solutions to see if there is something similar but for each solution I had seen I had doubts. I kindly ask you to check my solutions and ...

Question 8

A standard definition of an analytic set in $\mathbb R^n$ is a set $A$ such that there exists a Polish space $Y$ and a Borel set $B\subseteq \mathbb R^n\times Y$, such that $A = \pi(B)$, where $\pi:\...

Question 9

(Truncation). Let $f : \Omega \rightarrow \overline{\mathbb R}$ be a function. For a real number $M > 0$, we define the truncation of $f$ to be the function $f_M : \Omega \rightarrow \mathbb R$ ...

Question 10

Let $X\subseteq\mathbb{R}^n$ be a Borel set. Suppose $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension and $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on ...

Question 11

Lusin's theorem is generally stated for set of finite measure. In Royden's 'Real analysis' book the theorem is Lusin's Theorem Let $f$ be a real-valued measurable function on $E$ (Lebesgue-measurable ...

Question 12

Let $F_{1},F_{2},\dots$ be Borel‑measurable subsets of $\mathbb{R}^{2}$ whose union is the entire plane. Show that there exists an index $n\in\mathbb{N}$ and a circle $S\subset\mathbb{R}^{2}$ such ...

Question 13

Let $X,Y$ be Polish spaces and $f:X\to Y$ Borel measurable. Is $$Z = \{y\in f(X): f^{-1}(\{y\}) \text{ is countable}\}$$ a Borel set? If $Z=f(X)$ (i.e. all fibres are countable), then the result is ...

Question 14

I am looking for resources on fibres of continuous functions from $\mathbb R$ to $\mathbb R$. Here by fibres of a real valued function $f$, I mean the set $f^{-1}(y)$ for $y\in \mathbb R$. For context,...

Question 15

I think I'm being dumb but suppose that $P(f, g)$ for $f, g \in \omega^\omega$ is $\Pi^1_1$ and that $\forall f \exists g P(f, g)$. It should be true that for every $f$ there is a $g$ hyperarithmetic ...

Stack Exchange Network

Questions tagged [borel-sets]

Measure Theory - MIRA Book - Understanding the proof of 5.41

Borel sigma algebra of the range of an injective measurable function consists of all images of the Borel sets of the domain

When compacts generate the Borel $\sigma$-algebra, is $X$ $\sigma$-compact?

Characterizing Sets that Allow Paths to be Drawn in Mixed Rational/Irrational Dust

On the uniqueness of $\sigma$-finite measures

Measure on Extended Real Line

Given $\epsilon > 0$, there exists an open interval $I$ such tha $m(E \cap I) > m(I) - \epsilon$: solution verification

Definition of analytic set in $\mathbb R^n$

If I have a measurable truncation function $f_M$ of $f$, how do I prove $f$ is measurable as well?

Can all sets with infinite Hausdorff measure in its dimension be broken down into sets with positive measure?

Lusin's theorem on set of infinite measure

Dense Circle Intersection in a Countable Borel Covering of $\mathbb{R}^2$

Is the set of all points with countable pre-image a Borel set?

Fibres of continuous functions from $\mathbb R$ to $\mathbb R$ [closed]

$\Pi^1_1$ Uniformization by HYP (when total)

Hot Network Questions