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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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Prove $f=0$ almost everywhere with $L^p$ condition

Let $f:\mathbb{R}\to \mathbb{R}$ be a measurable function such that: 1) there exists $p\in (1,\infty)$ such that $f\in L^p(I)$ for all bounded interval $I$. 2) there exists $\theta \in (0,1)$ such that $$ \left| \int_I f\; dm \right|^p \leq \theta…
michaelaba
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Lebesgue-measurable subsets of $[0, 1]$

Let $A_1, A_2,\cdots$ be Lebesgue-measurable subsets of $[0, 1]$ of measure $1/2$, and let $A$ denote the set of points $x$ such that $x$ belongs to infinitely many of the sets $A_n$. How we can show that the Lebesgue measure of $A$ is at least…
Segni
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Sigma algebra generated by indicator functions on unit interval

I am reading the book Essentials of Probability Theory for Statisticians for self-study and I am stumped on the following exercise (p.47) 2. Let ${(\Omega,\mathcal F, P)}$ be the unit interval equipped with the Borel sets and Lebesgue measure. For…
user6997
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Measurable set invariant by rotation is radial

Let $A$ be a borelian subset of $\mathbb{R^n}$ contained in the unitary ball centered in 0. Assume that $m(A\cap Q(A)) = m(A)$ for each rotation $Q$, where $m$ is the Lebesgue measure. Show that there exists a radial function $f$ such that $\chi_A -…
d. zeffiro
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A problem about Lebesgue decomposition and Radon-Nikodym derivative

Let $f(x)$ be the Cantor-Lebesgue function. Write $$g(x)=x+f(x)$$ and let $\mathbb{B}$ be the sigma algebra generated by preimages of intervals under $g$, $\mathbb{C}$ be the sigma algebra which is the intersection of the algebra of Lebesgue…
Jack
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why does this simple function converge to f(x) pointwise

Hi: I'm reading some notes on measure theory and I don't understand one of the steps where the author defines the integral of f with respect to $\mu$. The link is here: http://www.martinorr.name/2008/probability/PM.pdf and the paragraph i am…
mark leeds
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Which measure gives this exterior measure $\mu^*$?

Let $\mu^*:\mathcal P(\mathbb R^2)\longrightarrow \mathbb R$ defined by $$\mu^*(E)=\inf\left\{\sum_{i=1}^\infty m(T_i)\mid E\subset \bigcup_{i=1}^\infty T_i\right\}$$ where $T_i$ are triangles and $m$ is the Lebesgue measure. 1) Show that $\mu^*$ is…
MSE
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Is the outer measure of $[0, 1] $ equal to $0$?

I am going to try and prove that the outer measure of $[0, 1] $ is $0$. I would be grateful if someone could point out the mistake. The outer measure of an interval is defined as $\inf \Sigma {l (I_n)} $ for any open covering of the interval $\{…
user67803
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If E has positive measure, does $E^2$ have positive measure?

I know the statement " If $E$ is a subset of real numbers with measure zero,then so is $E^2$={$x^2$: $x\in E$} " is true. How about " If $E$ is a subset of real numbers with positive measure, then so is $E^2$={$x^2$: $x\in E$} " ? Is true or false ?
C-HJ
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example of a prop of measurable functions

We proved in class that a function $f$ is lebesgue measurable if there exists a increasing sequence of simple functions that converge pointwise to $f$. That's OK but how can I build this sequence given the function? For example how could be a…
Mr. Leblanc
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Outer Measure of a countable union of disjoint intervals

If $E=\cup^{\infty}_{n=1} I_n$ is a countable union of pairwise disjoint intervals, I want to show that $m^*(E)=\sum^{\infty}_{n=1} m^*(I_n)$, where $$ m^*(E):=\{\sum^{\infty}_{n=1} Length(J_n): E\subset \cup^{\infty}_{n=1} J_n \} $$ each $J_n$ is a…
MAM
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Determining dominating function

Using lebesgue dominated convergence, calculate $\displaystyle \lim_{n\to \infty}\int_a^{\infty} n e^{-nx} \cos{x}\, dx$ when $a > 0$ and $a = 0$. I pretty much understand when $a>0$ since $ne^{-nx} \cos{x}$ is approaching $0$. But, there is no…
mohlee
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Lebesgue measurable subset of positive measure

Let $f:\mathbb{R}^d\rightarrow\mathbb{C-{0}}$ be Lebesgue measurable with $f(x+y)=f(x)f(y)$ for all $x,y \in \mathbb{R}^d$. Let $U\subset\mathbb{C}$ be a neighborhood of $f(0)=1$. I want to find some Lebesgue measurable subset $A$ of $\mathbb{R}^d$…
user193164
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Measure of irrational numbers on an bounded interval $[a,b]$

Since I am new to measure theory, I would like to ask the following question: I do know that the measure of the rational numbers $\mathbb{Q}$ in a real interval $[a,b]$ with $b>a$, has a measure of zero. But how can I prove that the measure of the…
Bazinga
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Construct a (non-measurable) function $f: R\to R$ with the following property

any function $g:R \to R$ such that $|g(x)-f(x)|<1$ for all $x \in R$ is non-measurable. I am thinking about the function $f$ being $f=2\chi_N$, $N$ is a non-measurable set in [0,1]. Thus, $m(x\in R, 1
user119758
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