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).

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pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ?
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Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
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Can someone show me why mathematicians use $d\mu$ instead of $dx$ for Lebesgue Integral over $u(x)$

I am an engineer and I learned my Lebesgue integral from an engineering text which dumbed down a lot of stuff, most prominently all Lebesgue integrals were introduced as $\int_\Omega u(x) dx$ instead ...
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1answer
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Is proving $m(E) < \epsilon, \forall \epsilon > 0$ equivalent to prove $m(E) = 0$?

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function ...
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if the integrals of a non-negative sequence of functions go to zero, does this imply functions go to zero a.e.? [duplicate]

This question arised when I was dealing with an old qual problem, and if this is true, I'll be done, but I'm not sure if it's true or not: Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of ...
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Show that there exists a $g \in L^1(m)$ s.t. $\phi (F(x))=\int_{0}^{x} g(t)dt$, where $F(x)=\int_{0}^{x} f(t)dt$

Let $m$ be Lebesgue measure on $[0,1]$ and suppose $f \in L^1(m)$ and let $F(x)=\int_{0}^{x} f(t)dt$. Suppose $\phi$ is a Lipschitz function. Show that there exists a $g \in L^1(m)$ s.t. $\phi ...
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Show that $\int_{\pi}^{\infty} \frac{1}{x^2 (\sin^2 x)^{1/3}} dx$ is finite.

Show that $\int_{\pi}^{\infty} \frac{1}{x^2 (\sin^2 x)^{1/3}} dx$ is finite. I've been trying to use Holder inequality but it seems I can't get the right combination of $p$ and $q$. Maybe I'm on the ...
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Find Lebesgue measure of $\limsup A_n \cap B_n$ if $m(\limsup A_n)=m(\liminf B_n)=1$,

Let $m$ be the lebesgue measure on $X=[0,1]$. if $m(\limsup\limits_{n\rightarrow{\infty}} {A_n})=1$ and $m(\liminf\limits_{n\rightarrow{\infty}} {B_n})=1$, prove that ...
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f:Lebesgue measurable function ⇆ ∀ε>0, ∃g:continuous function s.t. λ({x|f(x)≠g(x)})<ε

my friend told me this non-obvious prop. I think false,but I can't understand. Does anyone solve this problem?
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Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
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Explicit construction of a nonmeasurable set, where only the proof of correctness uses choice?

By Solovay's theorem, assuming the existence of an inaccessible cardinal, the axiom of choice is necessary to prove the existence of nonmeasurable sets. In the past, I've thought that one consequence ...
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68 views

Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...
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Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
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Compact $K\subset A$ such that $\lambda(K) = \lambda(A) / 2$

Let $A\subset \mathbb{R}$ be a (Lebesgue) measurable set of finite measure. Using the fact that the function $f:\mathbb{R}\rightarrow \mathbb{R}$, $$f(x)=\lambda(A\cap [-x,x]) $$ is continuous, we ...
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Transformation of a subset of compact Jordan sets to manifolds

Let $T$(for e.g. $[0,1]^2$) be a Jordan compact sets and $\tau$ be a "smooth enough one-to-one" transformation, i.e.($\tau: [0,1]^2 \rightarrow [0,1]^2 $). Lets take a subset of Lebesgue measure ...
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1answer
62 views

How to prove a set contains no rational numbers?

Let $E\subseteq \Bbb R$ be a set of Lebesgue measure zero. Show that there exists $a \in \Bbb R$ such that the set $$E+a :=\{x+a:x\in E\}$$ contain no rational numbers. I tried to use there is a ...
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30 views

Absolutely continuous function whose derivative is in $L^2([0,1])$ etc., evaluate $\lim_{x\to0^+}\frac{f(x)}{\sqrt{x}}$

Suppose $f$ is absolutely continuous on $[0,1]$, $f(0)=0$, and $f'(x)\in L^2([0,1])$. Show that $$\lim_{x\to 0^+}\frac{f(x)}{\sqrt{x}}=0$$ So far I've got the following: Since $f$ is AC by the ...
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1answer
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Relating Integration by Substitution to Change of Variables Theorem

I'm having trouble relating the change of variables theorem from measure theory to the integration by substitution formula taught in Calculus. I've always thought they were basically saying the same ...
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The weighted Signed measure

Given $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\mu$ be a finite signed measure. Then we know that, for $\varphi\in C_c^\infty$, $$ \sup_{\|\varphi\|_\infty\leq ...
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1answer
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Sequences of functions that converge uniformly, or pointwise, but not in $L^1.$

I'm reading the book Real Analysis of Folland. When I reached chapter 2 about the different modes of convergence, there's an example Folland gave that confused me: The 2 function sequences: ...
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Compute $\lim_{n\to\infty} \int_0^{\infty}\frac{e^{\frac{-x}{n}}}{1+(x-n)^2}dx$

Compute $\lim_{n\to\infty} \int_0^{\infty}\frac{e^{\frac{-x}{n}}}{1+(x-n)^2}dx$. I'm considering to use Dominated convergence theorem but the hint of this problem is the limit is non-zero. Any hints ...
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1answer
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$f\nu_{}=\big.m\big|_{[0,1]}$ where $m$ is the Lebesgue measurable.

If consider the map $f:2^{\omega}\to [0,1]$ given $f(x)=\sum_{i=0}^{n}x(i)2^{-i-1}$. Let $\nu_{}$ be the Haar measurable on $2^{\omega}$. Then $f\nu_{}=\big.m\big|_{[0,1]}$ where $m$ is the Lebesgue ...
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Absolutely continuous measure

If I have a measure $\mu$ on $[0,1]$ and if I know that $\int_{[0,1]}Gd\mu\leq\int_0^1|G(r)|dr\quad \forall G\in C[0,1]$ this implies that the measure $\mu$ is absolutely continuous with respect the ...
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Probability Triple [closed]

I'm preparing for my qualifying exam in 3 weeks time and came across this question in a text book. I just don't understand how to go by it even though I know the definition of Probability triple. So ...
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1answer
48 views

Lebesgue integral of Dirac delta

If I recall correctly, for a bounded function $f$ $$ \int_{\mathbb{R}} f \, d\mu = \int_{\mathbb{R} \setminus \{ a \} } f \, d\mu + f(a) \mu (a).$$ For the Lebesgue measure, $\mu(a) = 0$ and $$ ...
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Equivalent conditions of Lebesgue measurable sets

Hi I'd appreciate if someone can check the following exercise any suggestions are welcome. Thanks ;) Let $A$ a subset of ${\bf{R}}^d$ show that the following conditions are equivalent: (i) $A$ ...
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Relation between Lebesgue measure of two sets

question in lebesgue measure: Given that $T$ is a Jordan set of positive Lebesgue measure, $l(T)>0$. If $M \subset T $ such that $l(M)=0$ where $l(\cdot)$ denote Lebesgue measure, is it true ...
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Construction of a set with density of half at $0$.

If we define for a given set $A \subset \Bbb{R}$ and $x\in \Bbb{R}$ the density of $A$ at $x$ being the limit as $[I]$ goes to zero of the ratio $[I \cap A]/[I]$ wherever the limit exists for ...
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Small derivative and the measure of a set.

Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function, and that on some interval $(a,b)$, $|f'|\leq1$. Is it true that for all measurable sets $E\subset(a,b)$, ...
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Condition for $\overline{M}$-measurable in problem 2.24 by Folland

I'm self-learning Real Analysis using Real Analysis of Folland, and I got stuck on this problem. Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X) < \infty$, and let $(X, ...
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1answer
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If a function is Lebesgue measurable on each set, will it be measurable on its countable union?

Definition of measurable set: A set $E$ measurable if $$m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)$$ for every subset of $A$ of $\mathbb R^n$. Definition of Lebesgue measurable function: Given a ...
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Are all measure zero sets measurable?

Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} \ell(I_n): E \subset ...
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Volume of Manifold with zero Lebesgue measure

Let $M$ be a smooth manifold in $\mathbb{R}^n$. If Lebesgue measure of $M$ is zero i.e $l(M)=0$, does it mean that volume of manifold is also zero i.e $Vol(M)=0$? Are they the same thing (volume and ...
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1answer
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Prove continuity of averaging function for integrable $f$

I want to prove the following statement which is part of a lemma in my textbook: Suppose $f$ is integrable on $\mathbb{R}^n$ and $x$ be a lebesgue point of $f$. Let $$M(r)=\frac{1}{r^d}\int_{|y|\le ...
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Characterize the $\mu^*$- measurable sets where $\mu^∗ = \lambda^* \circ \text{proj}_1 $ and $ \lambda^*$ is the Lebesgue outer measure

Hi I'm working with Cohn's book and I have other problem with the necessity condition, I'd appreciate any help. Let $\lambda^*$ the Lebesgue outer measure on $\bf{R}$, and let $\pi$ be the ...
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1answer
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Is the product $d x \otimes \mu_x(d y)$ on $[0,1] \times [0,1]$ a probability measure?

Let $Y=[0,1]$ and $X=[0,1]$ (both with the usual Borel sigma algebra). Let $\mu_x$ be a probability measure over $Y=[0,1]$ for each $x \in X$. (i.e $\mu_x$ depends on $x \in X$). Let $d x$ be the ...
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Absolutely Continuous Weakly Convergent Sequence Need Not Converge Strongly

The following appears as an exercise in Sinai and Koralov's Theory of Probability and Random Processes. Give an example of a family of probability measures $P_{n}$ on ...
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Subsets $B$ of bounded subinterval $I$ is lebesgue measurable iff $\lambda^*(I)=\lambda^*(B)+\lambda^*(I\cap B^c)$

Hi I was reading Cohn's book and I have problem with the following exercises (only the return of b is what I don't know), I'd appreciate any help and suggestion, if necessary, for a): a) Show ...
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Product measurable set induced by nonnegative function

Suppose $(X, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space and $f: X \rightarrow \mathbb{[0, \infty]}$ a nonnegative function. Let $G_f = \{(x,y) \in X \times [0, \infty]: y \leq f(x)\}$. ...
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1answer
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Outer measure exclusion of zero set

I've just started self-studying measure theory by reading Pugh's Mathematical Analysis. He shows that the exclusion of a zero set does not change the outer measure: $m^*(E\setminus Z)=m^*(E)$, but ...
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If $A_1 \subset A_2 \subset \mathbb R$ and $m^*(A_1) = m^*(A_2)$, will $m^*(A_1 \cap T) = m^*(A_2 \cap T), \forall T \subset \mathbb R$?

Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} l(I_n): E \subset ...
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Prove that $\lim\limits_{n \to \infty } \int_0^\infty (1 + x/n)^{-n}x^{-1/n}dx= 1$ using DCT

Prove that $\mathop {\lim }\limits_{n \to \infty } \int_0^\infty {\frac{{dx}}{{{{(1 + \frac{x}{n})}^n}{x^{\frac{1}{n}}}}}} = 1$ using dominated convergence theorem (DCT). By DCT we need to show ...
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lebesgue measure basic exercise

I have a basic question about a Lebesgue measure exercise that I am not sure how to solve. (I apologize if this is a simple question, I am new with this subject). Compute the Lebesgue measure of $X$ ...
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for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
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Subset of Jordan set of positive lebesgue measure

let $T \subset \mathbb{R}^d$. Given on $T$, a Jordan set of positive Lebesgue measure, $l(T)>0$ . Let a set $M \subset T$; with $l(M)=0$. Please explain what is special about the set M. Has it got ...
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Lebesgue Measure of Image of Unit Square under Continuous Map

Problem. Let $h\in C(\mathbb{R})$ be a continuous function, and let $\Phi:\Omega:=[0,1]^{2}\rightarrow\mathbb{R}^{2}$ be the map defined by \begin{align*} ...
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Is $L^p(\Omega) = L^p(\bar {\Omega})$?

Is it true that $L^p(\Omega) = L^p(\bar {\Omega})$, where let us say $\Omega$ is a bounded domain of $\mathbb{R}^n$ with smooth boundary? I think it is true, because $\partial \bar {\Omega}$ has ...
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39 views

A weaker form of Lebesgue's differentiation theorem in $\Bbb R ^n$

If $f : \Bbb R ^n \to \Bbb C$ is locally-integrable then Lebesgue's differentiation theorem says that $$\lim \limits _{r \to 0} \frac 1 {\lambda \big( B(x, r) \big)} \int \limits _{B(x, r)} f \Bbb d ...
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1answer
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Dunford Pettis Theorem

Suppose $L_1([0,1],\lambda)=L_1(\lambda)$ is the set of all $1$-integrable functions on $[0,1]$. $$S=\{(f_1,f_2)\in L_1^2(\lambda) |0\leq f_1+f_2\leq 1, a.e. \}$$ By Dunford Pettis theorem, we know ...
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$\pi-\lambda$ Theorem to show measure giving interval lengths equivalent to Lebesgue on [0,1]

so I have been working on this problem and I want to make sure I am understanding the conclusion fully. So I have the following scenario: Not part of the actual question, but relevant. Consider ...