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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Prove that if $B = \{x-y : x,y \in A\}$, where $A$ is a Borel measurable subset of $R$ with positive measure

Suppose that $m$ is Lebesgue measure, and $A$ is a Borel measurable subset of $R$ with $m(A) > 0$. Prove that if $B = \{x + y : x,y \in A\}$, then $B$ contains a non-empty open interval centered at ...
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2answers
28 views

Are there two different notions of “conditional probability”?

This question comes from reading the discussion here. (1) If one is given a "probability measure" $P : F \rightarrow [0,1]$ mapping a Borel $\sigma$-algebra $F$ to $[0,1]$ then for two ``random ...
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1answer
18 views

Is the class of negligible sets a monotone class?

Let $(X,\Sigma,\mu)$ be a measure space we say that the subset $F$ of a set $X$ is negligible set if there exist $G \in\Sigma$ , $F$ is a subset of $G$ And $\mu(G)=0$ My question is: Is the class of ...
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1answer
23 views

Is a monotone function defined on any kind of interval measurable?

Definition of Measurable set: A set $E$ measurable if $$m^*(A) = m^*(A ∩ E) + m^*(A ∪ E^c)$$ for every subset of $A$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function $f: D ...
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Lebesgue-Stieltjes: Computation

Problem Given the real line $\mathbb{R}$. Consider a Borel family: $$\mu(\mathbb{R})<\infty:\quad\mu(\lambda):=\mu(-\infty,\lambda]$$ How can I compute: ...
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1answer
25 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
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28 views

How can I prove that $f$ and $g$ are measurable functions [on hold]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
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1answer
22 views

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 $$\lambda(O)=K?$$
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1answer
44 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
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2answers
45 views

Does there exist a subsequence whose intersection has measure greater than $1/2$?

I ran across the following problem on this review guide. It is problem 1.25, though I've changed the wording slightly. The measure is implicitly Lebesgue measure. Let $E_n$ be a sequence of ...
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57 views

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if ...
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1answer
29 views

Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
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55 views

A Ham Sandwich type problem

If $A_1,...,A_n$ are measurable subsets of $S^n$, then there is a great $S^{n-1}$ cutting each $A_i$ exactly in half. The tools I have at my disposal are the Borsuk Ulam theorem and the Ham ...
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1answer
29 views

Lebesgue integral of a ratio of Lebesgue densities

I need a hint to solve the following problem: $P$ is a probability mass on $\mathcal B(\mathbb R)$ with a Lebesgue density $h$, $f$ is another Lebesgue density. I need to show that $\int ...
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1answer
22 views

How does the Lebesgue measure measure non-cartesian product sets?

Consider the measure space $(\mathbb{R} \times \mathbb{R}, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda)$ where $\lambda$ is the Lebesgue measure on $(\mathbb{R}, \mathcal{B})$. Since ...
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1answer
43 views

Union of a null set and a non-measurable set

Suppose $S$ is a non-measurable set (wrt the Lebesgue measure) and $N$ has measure 0. What can be said about $S\cup N$? Is it also non-measurable?
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Lebesgue Integration Question

Let $f$ be integrable with respect to a Lebesgue measure. Evaluate the limit, $$\lim_{n \to \infty} \int_{-\infty}^{\infty} f(x-n)\left(\frac{1}{1+|x|}\right)\,dx$$ I tried change of variables ...
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1answer
52 views

Compute the following Lebesgue Integral

I've been sitting on this question for a while now: Let $f(x) \in L^1(\mathbb{R})$. Compute $$ \lim_{h\to \infty} \int_{\mathbb{R}} |f(x+h)-f(x)|dx. $$ I've managed to convince myself that the ...
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0answers
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A problem about a family of mesurable fuctions

Let $(X,\mathcal{M},\mu)$ a measure set such that $\mu(X)<\infty$ and $\mathcal{F}$ a family of $\mu$-measurable functions. Let $E(f,t)=\{x\in X\mid f(x)\geq t\}$ with $f\in\mathcal{F}$. If ...
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1answer
28 views

Interplay of Hausdorff metric and Lebesgue measure

Consider the space $\mathcal{K}(\mathbb{R}^{n})$ of compact subsets of $\mathbb{R}^{n}$ endowed with the Hausdorff metric $\rho$, and let $\lambda$ denote the $n$-dimensional Lebesgue measure on ...
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1answer
37 views

Integral of a nonnegative Lebesgue-measurable function on $ [0,1] $.

Let $ f $ be a nonnegative Lebesgue-measurable function on $ [0,1] $. Suppose that $ f $ is bounded above by $ 1 $ and that $ \displaystyle \int_{[0,1]} f = 1 $. Problem. Show that $ f(x) = 1 $ ...
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1answer
23 views

Partial Integration for measures

I have the following formula in mind, $\mu$ a measure on $\mathbb{R}$. Any sigma-finite measure on $\mathbb{R}$ can be decomposed into a absolut continuous part, a "point measure" and a singular ...
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1answer
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If$\mu$ is $\sigma$- finite, $\epsilon>0$, there exists $A\in \mathcal{A}$ such that $\mu(A)<\infty$ and $\epsilon+\int_A f>\int f$

Problem Let $X\mathcal{A},\mu$ be a $\sigma$-finite measure space. Suppose $f$ is non-negative and integrable. Prove that if $\epsilon>0$, there exists $A\in \mathcal{A}$ such that ...
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3answers
127 views

A function that is Lebesgue integrable but not measurable (not absurd obviously)

I think: A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$. However Royden & Fitzpatrick’s book "Real Analysis" (4th ...
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1answer
45 views

Unable to understand what kind of pdf and its origin

I am facing difficulty in identifying how the formula given by Eq(2) in the paper Wen-Chi Tsai and Anirban DasGupta, On the Strong Consistency, Weak Limits and Practical Performance of the ML ...
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1answer
46 views

What is the Lebesgue measure of a circle in $\mathbb{R}^2$

What is the Lebesgue measure of a circle in $\mathbb{R}^2$ So my answer of zero. The way I do it is by putting the circle inside an annular ring and then shrinking the ring so that its measure (which ...
6
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1answer
111 views

Does there exist a function $F(x)$, so that $F'(x) $ is not Riemann integrable?

Does there exist a function $F(x)$ such that it satisfies the following property? Let $I=[a,b]$ and $F:I\rightarrow\mathbb{R}$ be a function. $F$ is strictly monotonic and differentiable on $I$, ...
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Example: $|f|$ is integrable but $f$ is not integrable

Can someone give me an example about 1. A function $f$ that is not measurable but $|f|$ is measurable 2. A function $f$ that is not Lebesgue integrable but $|f|$ is Lebesgue integrable. (This ...
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Collection is uniformly integrable, but individual is not integrable

Could you give me an example about: "a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable." This sounds counterintuitive? However ...
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1answer
17 views

$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Let $a=\{a_i\}$ be an arbitrary sequence of complex numbers with finitely many non-zero terms. Consider the function $f(x)=\sum_ia_i g(x-i)$, where $g$ is a good function. Prove that for any $p\in ...
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1answer
55 views

Does measurability really matter?

I am studying applied math and I currently got stuck on proving that a function, which emerges in a model is measurable (Borel functon), so we can integrate it. I know, that there are examples of ...
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1answer
44 views

Find the limit $\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$

Problem Statement Find the limit $$\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$$ Attempt This problem is very similar to the following and I am basically going to ...
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0answers
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Construction of the Lebesgue measure

Is it possible to construct the Lebesgue measure on an $\underline{\text{abstract}}$ Boolean algebra? Or, does the greater abstraction of considering Boolean algebras instead of fields of sets ...
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30 views

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 ...
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1answer
40 views

$T:L^p \rightarrow L^p$ is bounded if it respects almost everywhere convergence

Let $T$ be a linear map from $L_p[0,1]$ to $L_p[0,1]$, $1\leq p < \infty$. If $(f_n)$ converges to $0$ almost everywhere, then $(T(f_n))$ converges to $0$ almost everywhere. How does this imply ...
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2answers
35 views

Lebesgue measurable and non-measurable sets

Is it possible to construct such $A \subset [0,1]^2$ that is not Lebesgue measurable, but it's projections to coordinate axes are measurable? Similarly construct subset that is measurable, but ...
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1answer
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A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable

Background Give an example of a collection of measurable non-negative functions $\{f_\alpha\}_{\alpha \in A}$ such that if $g$ is defined by $g(x)=\sup_{\alpha \in A} f_{\alpha}(x)$, then $g$ is ...
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1answer
35 views

Strictly Increasing function is measurable

I am having trouble proving the following questions: 1) Prove that if $f$ is strictly increasing on [a.b] then f is measurable (do not assume that $f$ is continuous). 2) Using the above, prove that ...
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Equivalence of definitions of measurable functions

Recently I read in a book that the definition $f^{-1}(E) \in X$ for every Borel set E is equivalent to the standard definition that f is measurable if for every real number $\alpha$ $\{x\in X: ...
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1answer
30 views

monotonic linear functional on $C_+(X)$

Let $X$ be a compact metric space. Let $C_{+}(X)$ be the set of all continuous non negative functions on $X$. Let $\lambda : C_{+}(X) \to [0,\infty)$ such that ...
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Measure of countable union

In my measure theory class, the professor prove that if $\mu$ is a finite measure on a space $X$ ($\mu(X) < \infty$) and $A_1 \subset A_2 \subset A_3 \subset \cdots$, then $\lim_{n\to\infty} ...
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A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Show that $\forall ...
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1answer
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Monotone convergence theorem in a special case

Suppose $C^{+}[0,1]$ be the set of all continuous functions with domain $[0,1]$ taking non-negative values only. Let $\lambda : C^{+}[0,1] \to [0,\infty)$ be a map that satisfies ...
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1answer
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Equivalences of weak convergence in $\mathcal{L}_p$ spaces with the Lebesgue measure

Let $\Omega =(0,1)$, and $f,f_n\in \mathcal{L}_p(\lambda)$. Prove that if $\sup_n{\| f_n \|}<\infty$ and $$\int_{(0,t]}f_n \, \,\mathrm{d}\lambda\rightarrow \int_{(0,t]}f \, ...
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Video lectures on Measure and Integration

Does anyone know a good online lecture series on measure theory and Lebesgue integration? I looked at the MIT open courseware but I could find only lecture notes. I am interested in lectures on this ...
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Lebesgue measure of a parallelepiped

Suppose we have $n$ linearly independent vectors $\mathbf{x}_1$, $\cdots$, $\mathbf{x}_n$ in $\mathbb{R}^n$. Let $\mathbf{X}$ be the $n \times n$ matrix with column $k$ given by $\mathbf{x}_k$, $k = ...
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1answer
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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 ...
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2answers
57 views

If $f_n (x)=\frac{n \sin x}{x (1+n^2 x^2)}$ then evaluate limit of integration $f_n(x)$ over $0 \to 1$ as $n \to \infty$

In this problem, I tried to dominated convergence theorem but I couldn't get any dominated function. How to find limit of this integration? Any hints or comments are welcomed.
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1answer
37 views

Is $\partial (A\times B)$ jordan measurable when both of $A$ and $B$ are jordan measurable?

If $A\subseteq \mathbb{R}^{n} $ is Jordan measurable, $B\subseteq \mathbb{R}^{m} $ is Jordan measurable, then $A \times B \subseteq \mathbb{R}^{n+m}$ is Jordan measurable? We have $$\partial ...
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1answer
54 views

Dual space of $L^{\infty}$ - Where is the mistake?

Today I thought about this for the first time and I really cannot see what is going on. I think it is a very stupid question but I really cannot see it. Consider the space $L^{\infty}(\mathbb{R})$ ...