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Questions tagged [lebesgue-measure]

For questions about the Lebesgue measure, a measure defined on the Borel or Lebesgue subsets of the real line or $\mathbb R^d$ for some integer $d$. Use it with (tag: measure-theory) tag and (if necessary) with (tag:lebesgue-integral).

Lebesgue measure is the classical notion of length and area to more complicated sets, and its assigns a measure to subsets of $n$-dimensional Euclidean space. Some examples of Lebesgue any closed interval, any cartesian product of intervals, any Borel set, and any countable set of real numbers (which has Lebesgue measure zero).

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If disjoint $S$ and $T$ are „locally“ either both null sets or not, can they have positive measure?

Let $S,T\subseteq \mathbb R^n$ be disjoint such that for every compact $K\subseteq \mathbb R^n$ we have that $K\cap S$ is a null set if and only if $K\cap T$ is a null set. Can both have positive Lebesgue measure? Intuitively, if one of them had…
Lukas Juhrich
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What's the reason for the pre-image def. of Lebesgue measurable?

What's the reason for the pre-image def. of Lebesgue measurable? I.e. $f$ Lebesgue measurable if $\{ x : f(x) > c \in \mathbb{R}\}$ is measurable. Like why is it significant that $f(x) > c$?
mavavilj
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How do I prove that counting measure is subadditive?

How do I prove that counting measure is subadditive? Particularly, I'm confused about, what cases do I need to explore? $$\mu^* \bigg(\bigcup_{i=1}^{\infty} E_i \bigg)$$ then since each $E_i$ is either some finite natural number or $\infty$, then do…
mavavilj
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If $A$ is lebesgue measurable, $B \subset A$ and $A-B$ is lebesgue measurable then B is measurable.

I am trying to prove this statement that seems very easy but i am struggling with it. If $B-A$ is measurable then $B^c\cup A$ is also measurable but that does not help really. I am pretty sure the end goal is to manipulate this to get $A$ by itself…
Sorfosh
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Can one construct a set $E$, $m^*(rE)\neq rm^*(E)$

Can one construct a set $E$, $m^*(rE)\neq rm^*(E)$. Well , we know that if $E$ is measurable then $$m^*(rE)= rm^*(E)$$,so what is the case of a none-measurable set ? Well,I think it holds for any set $E$, we assume that $$m^*(E)=\sum_n |I_n|$$ Where…
Alexander Lau
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Collection of all $T\subseteq\mathbb{R}$ such that $λ^{*}(T) =λ^{∗}(T∩A)$ for all $A\subseteq\mathbb{R}$.

Hi I am doing a past paper and I am stuck on the following question. Let $\theta$ be the collection of all $T\subseteq\mathbb{R}$ such that $λ^{*}(T) =λ^{∗}(T∩A)$ for all $A\subseteq\mathbb{R}$ , show that $\theta$ can be written as $\theta$ =…
RJSS
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In Lebesgue measure, why do we do outer approximations by open sets instead of closed sets?

So from Caratheodory's condition we can show that a set is measurable if it can be enclosed in an open set whose measure is equal / arbitrarily greater than the original set. What is the reason that we are only considering outer approximations by…
user49404
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Find Lebesgue Null Set A so that A-A is a Neighbourhood of zero

I try to solve the following question: Show that there is a Lebesgue null set A so that A-A is a neighbourhood of zero; where $A-A :=\{x-y|x,y\in A\}$. I really don't no how to do this. Thanks to everyone who can help.
user608381
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Lebesgue Measure of Cantor Set

For a presentation, I am learning about the Cantor Set and how it is homeomorphic to the p-adic numbers. I was reading section two of this paper. In it it states that the Cantor Set has a vanishing Lebesgue measure. Wikipedia says: Given a subset…
bkarthik
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Show that λ defined as λ(E)=μ(A∩E) is a measure on Ω for E in Ω

Let μ be a measure in Ω and A be afixed set in Ω .Then show that λ defined as λ(E)=μ(A∩E) is a measure on Ω for E in Ω. How to prove this?
Shammu
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Showing a function is lebesgue measurable

I am given that $(\Omega, \Sigma, \mu)$ is a measure space. Also I am given that $f$ is a nonnegattive $\Sigma$-measurable function on $\Omega$. Now let a function $g:\mathbb{R} \rightarrow[0,\infty]$ and $g(t)=\mu(\{f>t\})$ if $t\geq0$ and $g(t)=0$…
LanaDR
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Extension of Holder's Inequality

I found the proof of Generalization of Hölder's inequality from Wikipedia I get the other part, but I don't get why in case 1: we have the inequality $\Vert f_1f_2\cdots f_n \Vert_r \leq \Vert f_1f_2 \cdots f_{n-1} \Vert_r \Vert f_n \Vert_\infty…
Andy
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Cardinality of the continuum and Lebesgue measure of zero

How to construct a set with cardinality of continuum and a Lebesgue measure of zero? For instance, a set of all rational numbers within (0,1) is a countable set with Lebesgue measure of zero. What about continuum?
Stepan
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Finite Sets in the sense of Lebesgue Measure

Show that the sets $S(x)=\{(m,n):m\in\mathbb{Z},n\in\mathbb{N}^+,|x-\frac{m}{n}|\leq\frac{1}{n^3}\}$ where $x\in\mathbb{R}$ are finite for almost all $x\in\mathbb{R}$ in the sense of Lebesgue measure.
mathman
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