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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Transfer Lebesgue measure on $\mathbb{R}^2$ to $\mathbb{C}_{\infty}$

I've got a quick question. Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}^2$. I want to express the following "intuitive statement" mathematically: Since $\mathbb{R}^2\cup\{\infty\}\simeq ...
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1answer
11 views

Limit of translates of characteristic function

This might be silly, but what is a simple way of showing that given a characteristic function of a lebesgue measurable set in $\mathbb{R}$ then we have $\lim_{t \rightarrow 0} \chi (x-t) \rightarrow ...
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Computation of a Lebesgue-Stieltjes integral

I am asked to compute the integral $\int_{(0,3a]}x\,dF(x)$ with $a > 0$ where $$F(x) = \begin{cases} \pi & 0\leq x < a\\ 4+a-x & a\leq x < 2a \\ (x-2a)^2 & ...
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2answers
36 views

Continuity of a Lebesgue indefinite integral over unbounded interval

We know that if $f : [a,b] \rightarrow \mathbb{R}$ is Lebesgue-integrable, then $$ F(x) = \int_{a}^{x} f(t) dt $$ is continuous. But if $f : \mathbb{R} \rightarrow \mathbb{R}$ is ...
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1answer
30 views

Measure Theory, $\sigma$-algebra Folland Problem 23

I'm preparing for my exam. Can anyone help me in this matter, is confusing to me thank you very much.
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1answer
20 views

The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by $$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$ I need to ...
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1answer
29 views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
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1answer
69 views

“+”-Sets are measurable.

$A$ is a subset of $\mathbb{R}^2$ that for every $(x,y) \in A$ there is a $\delta >0$ that $(x-\delta , x+\delta) \times \{y\}$ and $\{x\} \times (y-\delta , y+\delta)$ are subsets of $A$. prove ...
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1answer
25 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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1answer
28 views

Nonmeasureable subset of ${\mathbb{R}}^2$ such that no three points are collinear?

I'm exploring the properties of sets in the plane that do not contain a set of three collinear points. In particular, I'm interested in the "largest" they can be. Things I know so far: Assuming the ...
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1answer
38 views

Prove or disprove $\nu(E)=\lambda(f(E))$ is a measure provided that $f$ is nondecreasing and satisfies the N-condition.

Suppose $f$ is a non-decreasing continuous function from $[a,b]$ to $\mathbb{R}$, and $\lambda$ is the Lebesgue measure in $\mathbb{R^1}$. Also, $f$ satisfies the property that $f$ maps Lebesgue ...
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0answers
22 views

Deciding if a measure is dominated by the Lebesgue measure

We define $X := \{0,1\}, \mu := \frac{1}{2} (\delta_0 + \delta_1)$ and $(\Omega, \mathcal{F},\mathbb{P}) : = \bigotimes_{n=1}^{\infty} \left( X, 2^X,\mu \right)$. For $\omega \in \Omega$ we denote the ...
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2answers
169 views

Lebesgue measure of sets of empty interior

There is a proposition to the effect that if the Lebesgue measure of a set is zero, then it is a set of empty interior. Does the converse also hold true, i.e. is the Lebesgue measure of any set of ...
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0answers
20 views

If the support of a function is contained in a Borel set, is the support of its derivative also contained there?

Let $f$ be a function such as $\operatorname{supp}(f)\subset Q$ where $Q$ belongs to the Borel $\sigma$-algebra on $\mathbb{R}^d$ Do we have $\operatorname{supp}(f')\subset Q$?
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3answers
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If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$. I just started learning about measure this week, so I don't know any theory about measure except the definition of outer measure ...
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50 views

Show that $(\mathcal{M},d)$ is complete metric space

Let $(\Bbb{R},\mathcal{M},\mu)$ be the Lebesgue measure space modulo the equivalence relation $A\sim B$ if $\mu(A\bigtriangleup B)=0$. Let $d(A,B)=\mu(A\bigtriangleup B)$. Show that $(\mathcal{M},d)$ ...
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1answer
49 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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1answer
38 views

Prove that measure of $A$ is $1$

Let $A\subset (0,1)$ be a Lebesgue measurable set and $\lambda>0$. Suppose that if $0\le a<b\le 1$ then $\mu(A\cap (a,b))\ge \lambda(b-a)$. Prove that $\mu(A)=1$. It is clear that $\lambda \le ...
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22 views

Lebesgue Measure of a set satisfying infinitely many solutions of this inequality

I am trying to find the following. Suppose that $\alpha_k > 0$, and $\sum \alpha_k < 0$. Let's consider the set $$A = \{x\in(0,1) | \hbox{the inequality} |x -{p \over q}| < {\alpha_q \over ...
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0answers
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For any $A \subseteq \mathbb{R}^d$, there exists a $G_\delta$ set $H \supseteq A$ such that for every measurable E, $|A \cap E|_e = |H \cap E|$

For any $A \subseteq \mathbb{R}^d$, there exists a $G_\delta$ set $H \supseteq A$ such that for every measurable E, $|A \cap E|_e = |H \cap E|$ I've done the case that $|A|_e < \infty$ using ...
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1answer
27 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
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29 views

Regarding the Lebesgue constant for interpolation

I have a question regarding Lebesgue constant $\Lambda_{n}\left(\boldsymbol{\chi}\right)$, with which the worst case error between an interpolant $p\left(\boldsymbol{x}\right)$ and the function which ...
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1answer
25 views

What can you say about union of two non measurable set. They are measurable or not?

What can you say about union of two non measurable set. They are measurable or not? Is it necessarily true?Thinking about α-cantor set I wonder if the complement of a non measurable set is a non ...
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2answers
56 views

Use Dominated convergence theorem to show that $f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3}$ is differentiable

Let $$f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3},$$ how can we show that f is differentiable everywhere by using the Lebesgue dominated convergence theorem? I know this theorem as saying ...
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0answers
62 views

$f$ is Riemann integrable iif the set of discontinuous points of $f$ has Lebesgue measure zero

This is a well known result in mathematics, but it's my first time attempting to prove it. I'm following the second book of Analysis from Folland. Below are the notations used and the theorem, from ...
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1answer
49 views

Convergence of $f_n(x) = 2^n \cdot F(2^n (x-a_n))$ with $F(x) = e^{-x^2}$ with different notions of convergence.

I had my measure theory exam this morning, and one exercise was the following: I really can't see a solution. During the semester, we talked about almost everywhere convergence, almost uniform ...
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1answer
20 views

Non-negative Lebesgue Measurable Functions/Determining Measure of a Particular Set

I'm having some difficulties trying to figure out where to even start with this problem: Let $f$, $g$ be non-negative, measurable functions on $\left[ 0,1 \right]$ such that $\int_0^1 f(x)dx=2$, ...
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1answer
28 views

Show that $\mu$ is absolutely continuous w.r.t. $\mathcal{L}$ and find $\frac{d\mu} {d\mathcal{L}}$

Let $\mu$ be the unique Borel measure on $\mathbb{R}$ satisfying $\mu((a,b])=\arctan b-\arctan a$. Show that for any $\mu$-measurable subset $E$ of $\mathbb{R}$, $\mathcal{L}(E)=0$ implies ...
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1answer
7 views

For every $E \subseteq \mathbb{R}^d$ with $0 < |E|_e < \infty$ and $0 < a < 1$ there exists a cube Q such that $|E \cap Q|_e \geq a|Q|$

For every $E \subseteq \mathbb{R}^d$ with $0 < |E|_e < \infty$ and $0 < a < 1$ there exists a cube Q such that $|E \cap Q|_e \geq a|Q|$ Here the exterior Lebesgue measure $|E|_e$ is ...
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1answer
44 views

Royden - section 4.2, page 73 - linearity

In Royden's "Real Analysis" on page 73, after the proof of linearity and monotonicity of the Lebesgue integral of simple functions, there's a little paragraph that says that this linearity shows that ...
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1answer
37 views

Lebesgue Measure in ${R}^m$

I am trying to solve the following problem. Problem Statement If $E \subset \mathrm{R}^m$ and $\lambda_m\left(E\right) >1 $, where $\lambda_m$ is the Lebesgue measure on ${R}^m$, then there are ...
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How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
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1answer
45 views

Prove that $\int_{[c,d]}|f(x,y)|d\mathcal{L}(y)<\infty$ for $\mathcal{L}$-almost all $x\in [a,b]$.

Suppose $f(x,y)$ is a Borel function on $\mathbb{R}^2$ which is in the $L^2$-space with respect to the $\mathcal{L}\times\mathcal{L}$. Prove the following: Given any finite rectangle ...
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2answers
26 views

Help with a Royden exercise of measure

I'm solving the exercise 12, of section 4 The General Lebesgue Integral from the Royden's book Real Analysis 3rd edition: Let $g$ be an integrable function on a set $E$ and suppose that $(f_n)$ is a ...
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1answer
155 views

Lebesgue point of density on $[0,1]$ and Dynkin's theorem

The problem defines a density point $x\in[0,1]$ for a Borel set $A\subset [0,1]$ if $$ \lim_{\varepsilon \rightarrow 0^+} \frac{\mu([x-\varepsilon,x+\varepsilon]\cap A)}{2\varepsilon}=1.$$Denote all ...
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1answer
43 views

Let A be a measurable set in R. Let B be all of it's densed points. is B necessarily open?

Let $A \subset \mathbb{R}$ be a measurable set. Define $B$: $$B =\left\{x\in \mathbb{R}: \lim \limits_{\epsilon \to 0^+} \frac{m([x-\epsilon, x+\epsilon]\bigcap A)}{2\epsilon} = 1\right\}$$ Is $B$ ...
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33 views

Can I apply measure theory in non-mathematics fields?

I am working in a field where researches try to get insight about a complex process. I will give an example to demonstrate this. Let's say, we are attempting to get the most efficient and cost ...
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1answer
55 views

Lebesgue measure in one dimension

Let $A$ be Lebesgue measurable and $0<\lambda(A)<\infty$. Let $\alpha\in(0,1)$. Prove that there exists an open interval $P$ such that: $$\lambda(A\cap P)\leq\alpha\lambda(P)$$ I found a proof ...
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2answers
49 views

Let $E\subseteq\mathbb{R}$ be a borel measurable set with $m(E)=0$ and $f(x)=x^{2}$. Is $m(f(E))=0$?

Let $E\subseteq\mathbb{R}$ be a Borel measurable set with $m(E)=0$ and $f(x)=x^{2}$. Is $m(f(E))=0$? I think it is true, but I do not know how to prove it. The only think I have got is that, if ...
3
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2answers
43 views

Is Lebesgue integral over interior equal to the integral over the whole set?

I have a measurable set $S\subset\mathbb{R}$ and a measurable function $f\colon\mathbb{R}\rightarrow \mathbb{R}$. Is it true that $$\int\limits_Sf(x)\, dx=\int\limits_{\operatorname{int}(S)}f(x)\,d ...
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1answer
38 views

Show that $\lim_{n\to\infty}\int_{n}^{2n}f_n(x)dx=0$ if $f_n\to f$ in $L^1((0,+\infty))$ with respect to Lebesgue measure

Let $f_n\to f$ in $L^1((0,+\infty))$ with respect to Lebesgue measure. I am asked to show that $$\lim_{n\to\infty}\int_{n}^{2n}f_n(x)dx=0$$ I think I could use dominated convergence for the sequence ...
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3answers
27 views

Proving Lebesgue measurability of Dirichlet-like functions

Dirichlet function $D:[0;1]\to\mathbb{R}$ is defined by $$ D(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \not\in \mathbb{Q} \end{cases}$$ We say that a function ...
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36 views

Translation- and linear transformation- invariance of Lebesgue measure, Rudin

I'm reading Rudin's Real and Complex Analysis and I'm puzzled about theorem 2.20, in which the Lebesgue measure is constructed via the Riesz representation theorem. Specifically, parts (c) and (e) of ...
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1answer
29 views

Limit of density points is a density point? [closed]

Let $\lambda(A)>0$ for some $A \subset \mathbb{R}^n$ and let $x_n \in A$ be a sequence of Lebesgue density points of $A$ with $x_n \to x \in int(closure(A))$. Must $x$ be a density point as well?
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1answer
32 views

Question about Royden's proof about countable subaddivity of lebesuge outer measure

I have a question about the following proof by Royden. What I do not understand is the part where it says $\Sigma \Sigma l(I_{k,i}) < \Sigma m^*(E_k)+\epsilon / 2^k$ Why is there a strict ...
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1answer
46 views

Measure of the reciprocal of a Cantor set

I have recently started studying measure theory and as is usual we started out by calculating the measure of the Cantor set. Now I had this question in my mind as to whether the set generated by ...
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27 views

Is my proof of a dominated convergence corollary using Egorov's theorem right?

Theorem: If $\mu(\Omega) < \infty$ and the $f_n$ are uniformly bounded, then $f_n \to f$ almost everywhere implies $\int f_n d\mu \to \int fd\mu$. This is a simple consequence of Lebesgue ...
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Uniform Distibution Over Middle-Third Cantor Set (and Its Approximates)

Above is my question. I have done part 1, but I am unsure how to do the next part(s). Part 2 at least doesn't look that difficult, but I can't get the specifics. My main issue is that I can't get a ...
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0answers
20 views

Why do we construct the Lebesgue measure with finite measure sets before sets of arbitrary measure? [duplicate]

On page 20 of the following lecture notes, Stage 5 constructs the Lebesgue measure on finite sets before constructing it on arbitrary sets as in Stage 6: ...
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1answer
38 views

L-p space: p-norm proof

Can somebody put me in the right direction to prove that: $\lim_{p \to 1} \lVert f \rVert_{p}^p=\lVert f \rVert_{1}$ ? Maybe this will be a beginning: If $f \in$ $\mathcal{L}^1(\mu)\cap ...