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Questions tagged [measurable-sets]

For questions about measurability, that is, whether a subset of a general space belongs to the σ-algebra, or about properties of measurable sets. Use this tag with (measure-theory), (real-analysis), (probability-theory) or (geometric-measure-theory).

Intuitively, a measurable set is a set which can be assigned a meaningful "size" (formally known as "measure"). The notion of measurability allows formal definition of length, and in , and probability of events in .

In formal settings, a subset $E$ of the general space $\Omega$ equipped with a $\sigma$-algebra ${\cal F}$ is called measurable if $E \in {\cal F}$.

Remark: Note that the general space $\Omega$ does not need to have, a priori, a topological structure.

Reference:

  1. Wikipedia
  2. Wolfram MathWorld
275 questions
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What is the definition of a measurable set?

I have seen multiple definitions for what a measurable set is (all of which come together to form a sigma algebra). I was wondering if they are all equivalent and if not what situation would one be used over another? Definition 1 Let $(X,…
DanZimm
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Intuition behind the Caratheodory’s Criterion of a measurable set

This week I saw the definition of a measurable set for an outer measure. Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + \mu^*(A^c\cap E)$$ for every $E \subseteq X$. This is not…
Michael Albanese
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Definitions of Measurability: Outer-inner measure convergence vs. Caratheodory criterion

If we are looking over subsets of $\mathbb R$ and considering the outer measure defined exactly as $$\mu^*(A) = \inf\left\{ \sum_{k=1}^\infty \ell(I_k) \text{ where the $I_k$ are open intervals such that } A\subset \bigcup_{k=1}^\infty I_k…
D.R.
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2 answers

Absolutely continuous maps measurable sets to measurable sets

Show that if $f:\mathbb{R} \rightarrow \mathbb{R}$ is absolutely continuous, then $f$ maps measurable sets to measurable sets. Any ideas on how to do this?
Jake Casey
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11
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3 answers

Proving the *Caratheodory Criterion* for *Lebesgue Measurability*

I'm following Terry Tao's An introduction to measure theory. Here he has defined lebesgue measurability as: Definition 1 (Lebesgue measurability): A set $E$ is said to be lebesgue measurable if for every $\epsilon > 0$, $\exists$ an open set $U$,…
user399978
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1 answer

Is the continuous image of a Borel subset Lebesgue measurable?

Let $f:\mathbb R^n \to \mathbb R^n$ be a continuous map and $B \subset \mathbb R^n$ a Borel subset. It is well known that $f(B)$ may not be a Borel subset. My question is, can we prove that $f(B)$ must be measurable for Lebesgue measure?
Summer
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Proving Caratheodory measurability if and only if the measure of a set summed with the measure of its complement is the measure of the whole space.

Suppose we have a premeasure $\mu$ on a space $X$ such that $\mu(X) < \infty$. Prove that $E \subset X$ is Caratheodory measurable iff $ \mu^*(E)+ \mu^*(E^C) = \mu(X)$. Going in the forward direction is fairly easy. Assuming that E is Caratheodory…
sjgandhi2312
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Is it possible to redefine measurable spaces such that measurable functions are not defined in terms of pre-image?

We know that a measurable space is defined as a tuple $(X,\mathcal A)$, where $X$ is a set and $\mathcal A$ is a $\sigma$-algebra defined on $X$. A $\sigma$-algebra is traditionally defined as a collection of subsets such that the whole set is in…
HackR
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Lipshitz function and measurable sets

I apologize if that question has been asked already. I can't figure out this problem: Let a function $f:[a,b]\rightarrow \mathbb{R}$ which be a Lipshitz function with a constant c, prove that it maps a set of Lebesgue measure zero onto a set of…
Ilia
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2 answers

Existence of a subsequence in a set of positive measure for any sequence in [0,1]

The following is an exercise from Bruckner's Real Analysis: 5:12.2 Let $E$ be a Lebesgue measurable set of positive measure, and let ${\{x_n}\}$ be any sequence of points in the interval $[0, 1]$. Show that there must exist a point $y$ and a…
user200918
6
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1 answer

Measurable sets whose sum is not measurable.

I'm looking for two measurable sets in $\Bbb{R}^2$ st their sum is not measurable. I found the example , let $A\subset \Bbb{R}$ be a non-measurable set in $\Bbb{R}$ and consider the sets $A\times \{0\}$ and $\{0\}\times \mathbb R$, these both sets…
Mathronaut
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5
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commonness of non measurable sets

Many times in math the pathological examples are the standard and the nice examples are the exceptions to the rule, like differentiable functions within continuous ones, continuous ones within all functions, algebraic numbers within reals, principal…
Tychus Findlay
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1 answer

Is it possible to write a metric a space as a countable disjoint union of compact sets?

Let $ (X,d)$ be a metric space and let $\mu $ be a Radon $\sigma$-finite measure on the Borel $\sigma$-algebra. I read that it's possible to find countable disjoint compact sets $\lbrace K_n\rbrace_{\mathbb{N}}$ and a $\mu$-null set $N$ such that $$…
Emanuele Angilè
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Measurable functions : $f(A) \in \mathcal{B}$

I am new to measure theory and here is the definition I have : (1) A function $f:(X, \mathcal{A}) \to (Y, \mathcal{B})$ is mesurable iff : $\forall B \in \mathcal{B}, f^{-1}(B) \in \mathcal{A}$ Why this definition and not this one ? (2) A…
NickolasB
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Proving that $m^*(E)=\sup\{m(F)\mid F\text{ is closed}, F\subset E\}$ iff $E$ is measurable

My question is: Let $E \subset [0,1]$ be a set on $[0,1]$ and let $S=\sup\{m(F)\mid F$ is closed and $F \subset E\}$. Prove that $S=m^*(E)$ if and only if $E$ is measurable. I have not found any help here or on other websites for my specific…
heather
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