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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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About two measurable set question

Question: If $A$ and $B$ are measurable set, $ m(A) + m(B) = m(A\cup B) + m(A\cap B)$ equality is provided.(Lebesgue Measure) My Solution: $m(B) = m(B\cap A) + m(B \cap A^\mathbb{c}) = m(B\cap A) + m(B \setminus A)$ $m(A \cup B ) = m((A \cup B)…
Music Boy
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How to check Lebesgue Measurability of the given Set?

Let E be a measurable set. $x$,$y$ $\in E$ are $\delta$ equivalent if $x=2^{n}y$ for some integer $n$. The $\delta$-index of a point $x$ in $E$ is the number of elements in its $\delta$ equivalent class and is denoted by $\delta{_{E}} (x)$. Let…
Sunder Deep
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Why Borel measurable sets are Lebesgue measurable

Let K be an open set of $\mathbb{R}^n$, then I know that I can get a countable cover by using closed cuboids $Q_i$ with a pairwise disjoint interior. I want to show that by this fact every open set is measurable under the Lebegues measure. I…
Rico1990
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Subset of $\Bbb R$ that doesn’t satisfy $\lim\limits_{\epsilon\rightarrow 0}\mu((E+\epsilon)\backslash E)=0$

There is a subset $E$ of $\Bbb R$ that doesn’t satisfy $\lim\limits_{\epsilon\rightarrow 0}\mu((E+\epsilon)\backslash E)=0$ where $\mu$ is the Lebesgue measure. I came up with $\Bbb R\backslash \Bbb Q$ and $\bigcup\limits_{n\in\Bbb N}[n,n+{1\over…
John Cataldo
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Is there a collection of measurable not necessarily nested but converging sets such that we cannot permute measure and limit

We know that for $\{E_n\}_{n\geq0}$ Lebesgue measurable sets s.t. $E_0\subset E_1\subset...$ we have $\lim\limits_{n\rightarrow\infty}E_n=\bigcup\limits_{n=0}^{\infty}E_n$ and $\mu(\lim E_n)=\lim(\mu(E_n))$ Similarily if $E_0\supset E_1\supset...$…
John Cataldo
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compute the measure $ \large \ m_{*} (\mathbb{Q}^d \cap [0,1]^d) \ $

Let $ \ m_{*} \ $ be the Lebesgue Exterior measure on $ \ \mathbb{R}^{d} \ $ , where $ \ \mathbb{R}^{d} \ $ is the countable product of $ \ \mathbb{R} \ $. Then compute (i) $ \large \ m_{*} (\mathbb{Q}^d \cap [0,1]^d) \ $ Answer: If $ \ d \ $ be…
MAS
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What properties can be said for a subset of the reals which has a Lebesgue measure strictly greater than zero?

What properties can be said for a subset of the reals which has a Lebesgue measure strictly greater than zero? I tried googling but there weren't that many and if there was they were poorly explained
dahaka5
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How to use Lebesgue measure formula

I'm struggling to see how to use the Lebesgue measure formula. So for example given any subset $[a,b]$ of the reals, what is it's "length" according to the formula: $\lambda(A)=\inf({\sum|b_i-a_i|:\text{A subset of the union}\ [a_i, b_i] })$
dahaka5
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Hyperboloid with Lebesgue measure zero

Consider the surface (hyperboloid) $S=\{(x,y,z)\in \mathbb{R}^3: \frac{1}{z}+\frac{1}{y}+\frac{1}{x}=0\}$. How to prove that: it is of Lebesgue measure zero? The idea is to show that for all $\epsilon>0$, there is a countable set of Cubes$E_n$ such…
Raafat
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Calculate the Lebesgue integral

Calculate the Lebesgue integral $$ \int_{[0,\pi]}^{} \sin x\: \epsilon[\mathbb{R}\setminus\mathbb{Q}] d\mu,$$ where is $\epsilon $ is the characteristic function. I tried this way $ \int_{\mathsf ([0,\pi ] \cap I) }^{} \sin x \: d\mu$... $\mathsf…
Martin
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regularity theorem for lebesgue measure

I'm having trouble with this problem. I was able to show that if a set A is measurable then for every $\epsilon$ $>$ $0$, there exists both an open set $G$ and a closed set $F$ such that $F$ $⊂$ $A$ $⊂$ $G$ and $m($G\F) $<$ $\epsilon$. But I can't…
MathNoob
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Find out the outer measure of E and also show that it is measurable.

# Let $ E $ be the set of all real numbers in $ (0,1)$ such that each $ x $ has decimal representations with digits a $ 2 $ or a $ 4$ . Find out the outer measure of the set $ E $ and show that $ E $ is measurable set.
MAS
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example of measurable set in the plane which is not almost a countable union of rectangles

Does there exist a Lebesgue measurable set $E$ in the plane $\Bbb{R}^2$ such that for every countable sequence $(Q_n)_{n \in \Bbb{N}}$ of rectangles with sides parallel to the coordinates axes the Lebesgue measure of the symmetric difference $E…
Damiano Foschi
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Is $\lim_{h \to 0} \int_0^h |f(x+h)-f(x)| dx = 0$?

This question was mistakenly asked in $\mathrm{lim}_{h\rightarrow 0} \int_0^h |f(x+h)-f(x)| dx=0$ almost everywhere. But it turned out that the person was only asking for $$\lim_{h \to 0} \int_0^h |f(x+t)-f(x)| dx = 0$$ which is easy to verify. But…
Adam
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Is there a subsequence of sets $\{A_{k_j}\}$ where the intersection is positive.

Suppose $\{A_k\}_{k=1}^\infty$ a family of open subsets of $\mathbb{R}^n$ where $$ A_k\subset B_1=\{x\in \mathbb{R}^n : |x|<1\}\,\,\,\mbox{and}\,\,\,\mu(A_k)\geq\epsilon>0. $$ Is there some subsequence …
ITS
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