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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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Almost every point in a union A of measurable sets $A_n$ belongs to at most k of the sets. Show $m(A)\geq \frac{\sum m(A_n)}{k}$

Let $\{A_n\}$ be a collection of measurable sets in $\mathbb{R}$. If $A$ is the union of the collection and almost every $x\in A$ belongs to no more than $k$ of the $A_n$ then I need to show that $m(A)\geq\frac{\sum m(A_n)}{k}$. My idea: For each…
dsimo04
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Borel and Lebesgue measurable sets

How can i prove that the cardinality of Borel sets is less than the cardinality of lebesgue measurable sets?
Marios Gretsas
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How to solve Lebesgue on a set

First I'll have to mention this is not homework or something like that. It's training for an exam and I couldn't find resources on how it is done. If you have any URL showing how to solve these kinds of Lebesgue exercises, please tell me. I have the…
Tyraxor
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Proof the Lemma for use Jordan's Theorem

Show the Lemma following: Let the function $f$ be of bounded variation on the closed, bounded interval $[a,b]$. Then $f$ has the following explicit expression as the difference of two increasing functions on…
Julio_fmat
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If A is measurable, there exist F $\subset$ A and open set G s.t. A $\subset$ G s.t. $\forall \epsilon > 0$, $\lambda (G\setminus F) < \epsilon$.

Why my logic is flawed in this reasoning:- If A is measurable we have Sup $\lambda (F)$ = inf $\lambda (G)$ for all such G and F, therefore we can always find such F and G which are $\epsilon$ / 2 close to $\lambda (A)$.
canseeker
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How to show $m(A) = m(A \backslash Z)$

Let $m$ be the Lebesgue measure, and $Z$ a set of measure zero, $A \subset \mathbb{R}$ Then intuitively, $m(A) = m(A \backslash Z)$ How to show this? Attempt: $m(A \backslash Z) = m(A \cap Z^c)$ Is there a way to turn that $\cap$ upside down and…
Fraïssé
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n sets such that any x belongs to at least k implies that at least one set has measure at least k/n

Let $k, n \in \mathbb{N}: k ≤ n$ and let $E_1, \dots , E_n \subseteq [0, 1]: ((\forall i, E_i$ is Lebesgue measurable subset$)$ & $( x \in [0, 1] \Rightarrow |\{E_i: x \in E_i, i \leq n \}| \geq k))$; (that is: $x$ belongs to at least $k$ of the…
user332420
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Difference set for E measurable

Let $E\subset \mathbb {R^n}$ measurable (Lebesgue) such that $\mu(E)>0$. Prove that $D(E)=\{x-y:x,y\in E\}$ contains a ball centered in $0$. Any hint, please? Thanks!
Martín Vacas Vignolo
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Another Lebesgue measure question

Let $\mu$ be a Lebesgue measure on the Borel $\sigma$ algebra. Then is $\mu( [0,\frac{1}{4}) \bigcup [\frac{3}{4},1])$ just $\mu([\frac{3}{4},1]) + \mu([\frac{3}{4},1])$ with $\mu( [0,\frac{1}{4})) = \frac{1}{4} - 0 = \frac{1}{4}$ and…
josh
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Showing a linear properties of outer measure(Lebesgue).

For a subset $A$ of $\Bbb R$ and real numbers $a$ and $b$ define the set $$aA+b=\{ax+b:x\in A\}$$ Show that $m^{*}(aA+b)=|a|m^{*}(A)$ and if $A$ is Lebesgue measurable so is $aA+b$. I don't know how to show the first except this Since outer measure…
marya
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An exemple of measurable function $f$ on $\mathbb R^2$ s.t. $f^y$ is not mesurable for every $y$.

I'm looking for a function $f:\mathbb R^2\longrightarrow \mathbb R$ which is measurable but such $f^{y}$ defined by $f^{y}(x)=f(x,y)$ is not measurable for every $y$. Does $$f(x,y)=\chi_{\mathcal N\times \{0\}}(x,y)$$ where $\mathcal N$ is a non…
user301068
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If $f$ measurable, prove or disprove that $E=\{(x,\alpha)\mid 0\leq \alpha< |f(x)|\}$ is measurable.

Let $f:\mathbb R\to\mathbb R$ a measurable function. prove or disprove that $E=\{(x,\alpha)\mid 0\leq \alpha< |f(x)|\}$ is measurable. I know it's measurable, but I really have no idea how to prove it. P.S: Does $E=\{(x,\alpha)\mid 0\leq…
Rick
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Convergence of $L^{p}$ norm as function of $p$.

One can prove that in any measure space, if $\{p\in[1,\infty): \|f\|_{L^{p}}<\infty\}$ contains of two points, then this set is actually a connected subset of $\bf{R}$. Assume that it is of the form $[1,b]$, where $b<\infty$, can we have…
user284331
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Subset of null set (set of measure zero)

Is there a proper subset of a set of measure zero that is not measurable? Any examples? Thanks a lot! I suspect the answer is yes due to some careful phrasing in books, e.g. let F be a subset of a null set in A. If F were always measurable, it would…
yoyostein
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Lebesgue measurable function

I'm reading an article and have to use the following statement. If $g$ is a real valued function on $\mathbb{R}^{2}$ such that $g_{x}$ is Lebesgue measurable for all $x \in E$ and $g^{y}$ is continuos for all $y \in \mathbb{R}$ then $g$ is Lebesgue…
Craneviter
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