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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).

7551 questions
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Intersection of Borel sets with positive Lebesgue-Borel measure

Let $A \in \mathfrak{B}(\mathbb{R})$ be a Borel set with $\lambda(A) > 0.$ Are there $B, C \in \mathfrak{B}(\mathbb{R})$ with $ B\cap C =\emptyset, B\cup C = A$ and $\lambda(B), \lambda(C) > 0?$ Thank you.
Obriareos
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The (non)existence of a measurable set

Does there exist a measurable set $E \subset [0,1]$, such that for any open interval $I \subset [0,1]$, $$m(E~\cap~I)=\frac{1}{2}m(I)~?$$ Here $m$ denotes the Lebesgue measure.
AaronS
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Any measurable set has a closed subset such that the remainder has small outer measure

Let $A \subseteq \mathbb{R}$ be Lebesgue measurable. Prove that: $\forall ε > 0, \exists F_ε ⊆ A: F_ε$ is closed in $\mathbb {R},$ & $λ^{∗} (A \smallsetminus F_ε) < ε$. I have proven two similar results for an open subset version of this statement,…
user332420
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why do we need $f$ bounded for uniform convergence?

Theorem If $f$ is a bounded measurable function, then there exists a sequence of simple functions $\{f_{n}\}$ which converge uniformly to $f$. Proof Define $f_{n}(x) = \frac{m}{n}$ whenever $\frac{m}{n} \leq f(x) < \frac{m+1}{n}$. We clearly have…
Kashif
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Lebesgue outer measure of $\mathbb{R}$ in $\mathbb{R}^2$ is zero

Following Lebesgue measure of any line in $\mathbb{R^2}$. I want to show that the lebesgue outer measure of $\mathbb{R}$ in $\mathbb{R}^2$ is zero. Does the following constitue a good way to show this fact? Take $R_n = [n - \dfrac{1}{2} , n +…
Sarah Palins Anger
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borel outer measure on R^n is additive on sets with distance greater than 0.

Can anyone help me with the following question? Thank you very much in advance :) Let $u^*$ be a Borel measure on ${\mathbb R}^{n}.$ Show that $u^{*}(A \cup B) = u^{*}(A) + u^{*}(B)$ whenever $A$, $B$ are subsets of ${\mathbb R}^{n}$ with…
Measure Theorist
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what would be $F=\bigcup_{i=1}^N [a_i,b_i]$ in $m(E\Delta F)\leq \varepsilon$ if $E=\mathbb Q$?

Refer to this question, what would be such an $F$ if $E=\mathbb Q$ ? I mean, $\mathbb Q$ has measure null and thus the measure is finite. Therefore, for all $\varepsilon>0$, there exist $[a_1,b_1],...,[a_N,b_N]$ s.t. $$m\left(E\Delta \bigcup_{i=1}^N…
idm
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Show that the graph of a function is measurable.

Let $f:\mathbb R\to\mathbb R$ a measurable functionI have to show that $\Gamma=\{(x,f(x))\mid x\in\mathbb R\}$ is measurable and that $m(\Gamma)=0$. (I work with Lebesgue measure). My attempt If $f=1_F$ where $f$ is an interval, we have that…
idm
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Does $m^*(A\cup B)=m^*(A)+m^*(B)$

Let $m^*$ the Lebesgue exterior measure. We have by certain observation that : 1) if $E=E_1\cup E_2$ and $d(E_1,E_2)>0$ then $$m^*(E)=m^*(E_1)+m^*(E_2)$$ 2) If a set $E$ is the countable union of almost disjoint cubes $E=\bigcup_{i=1}^\infty Q_i$,…
Rick
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Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?

So for given $x \in [0,1]$, if we let $f_n(x)$ be the fraction of 1's occurring in the first $n$ binary digits of the binary expansion for $x$ (where we always assume an infinite trailing string of 0 instead of 1 for certain rationals $x$), then the…
user2566092
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Clarification on the definition of Lebesgue measure

I'm doing some independent reading on the Lebesgue measure and I have the following questions: 1) on the definition of a Lebesgue measurable set: Let $E \subseteq \mathbb{R} $. On Wikipedia, $$\mu(E) = \mu^*(E) \iff \mu^*(A) = \mu^*(A\cap E) +…
Kevin Sheng
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Why is the lebesgue measure equal to K?

I have the following question. Let $\lambda$ be the lebesgue measure. Let $O$ be open and dense in $[0,K]$. Why is $$\lambda(O)=K?$$
user252409
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Convergence pointwise but not in measure

Let $X=\mathbb R$ and $\mu=m$. Let $f_n(x)=e^{-|x-n|}$ and $f(x)=0$, $x\in\mathbb R$. Show that $f_n$ converges pointwise to $f$, but $f_n$ does not converge in measure to $f$. I didn't have any trouble showing $f_n$ converges pointwise, but my idea…
Kevin
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A simple question about Haar measure

$G$ is a locally compact Hausdorff topological (multiplicative) group, $m$ is a (left) Haar measure on $G$. I have known that for any $g\in{G}$, $m(gB)=m(B)$. My question is, for any Borel measurable set $B$, $m(B)>0$, can we conclude that…
David Chan
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Difference between Lesbegue measure and Borel measure

the Lesbegue measure is defined an the Borel Sigma-Algebra, so does the Borel measure. Can somebody point out the difference betweem them and give an example to observe that difference ? Thanks.
ivo
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