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).

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1answer
15 views

Fatou's Lemma applied to simple functions

Show that the sequence of measurable functions $f_i: \mathbb{R} \rightarrow \mathbb{R}$ defined via \begin{align*}f_i(x)= \begin{array}{cc} \{ & \begin{array}{cc} -1 & i \leq ...
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0answers
31 views

Convergence of $e^{iu_nx}, x \in \mathbb{R}$

Let $(u_n)_n$ be a sequence of real numbers. Suppose that $e^{ixu_n}$ converges to a limit for every $x$ in a set $K$ of strictly positive Lebesgue measure. So, is it true that there exists $\eta>0,...
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46 views

Papa Rudin exercise 2.8 [duplicate]

I am trying to solve Exercise 8, Chapter 2 of Rudin's Real and Complex Analysis, but I am not sure of what I have done and I am not able to answer to the final part of the question. The requirement is ...
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1answer
18 views

Show that there exists a subsequence $\{E_{n_k}\}$ of $\{E_n\}$ such that $m(\cap_{k=1}^\infty E_{n_k})>\epsilon$ under these conditions…

Question: Let $\{E_n\}$ be a sequence of nonempty Lebesgue measurable subsets of $[0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n)=1$. Show that for each $0<\epsilon<1$ there exists a ...
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1answer
50 views

Measure of $\varepsilon$- fattening

Let $E\subset\mathbb{R}^{n}$ be a bounded set and let $$ B(E,\varepsilon)=\{x\in\mathbb{R}^{n}:\text{dist}(x,E)\leq \varepsilon\}. $$ Is it true that $$ |B(E,\varepsilon)\setminus E)|\leq C\...
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1answer
25 views

Prove that $F$ is Lebesgue measurable and $\sum_{n=1}^\infty m(E_n)\geq Km(F)$ under these conditions…

Question: Suppose $E_n$, $n\in\mathbb{N}$, is a sequence of Lebesgue measurable subsets of $[0,1]$. Let $F$ be the set of all points $x\in[0,1]$ that belong to at least $K$ (some positive number) of ...
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3answers
54 views

Convergence of Lebesgue measurable sets

I've been working on the following result: Let $f$ be Lebesgue measurable on $[0,1]$ with $f(x)>0$ almost everywhere on $[0,1]$. Assume there are measurable sets $E_k \in [0,1]$ with $\int_{E_k} f(...
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1answer
33 views

Signed Measure $\nu$ mapping $A \mapsto \int_A f d\mu$ and the Radon-Nikodym derivative

I am supposed to take an exam in August and so I am trying to prepare (this is not homework). So far I am pretty good at most topics, but anything related to Radon-Nikodym I don't quite understand. ...
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1answer
19 views

Approximation of Dirac measure by absolutely continuous measures (w.r.t. Lebesgue measure)

Let $\mathcal{P}$ be the set of probability measures on the Borel $\sigma$ Algebra on the unit circle $S^1$. Let $a\in S^1$ and $\delta_a$ the Diracmeasure in $a$. I want to find a sequence $(\mu_n)_{...
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set of sequence with positive measure

This related to the question Binary number and measure. Consider the set of $\{x=(0.x_{1}x_{2}...)|x_{i}\in\mathbb{Z}_{2},i\in\mathbb{N}\}$, this is the set of all number in $[0,1]$, so it have ...
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27 views

Let $A$ be the event that the share price never moves up. Find, with a justification, the probability of $A$.

In the share price model which is a random experiment depicting the evolution of share price at random fixed time intervals where the probability of the price going up or down at time $t=n$ where n is ...
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24 views

The event that the share price never moves up three times in a row

In the share price movement example, we let $X = [ω = ω_1, ω_2, \ldots \mid ω_n = u \text{ or } d]$ and we let $E_n$ be the event that the share price moves up at time $t = n$. Write down a formula in ...
4
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1answer
85 views

Problem with showing $\lim_{n\rightarrow \infty} \int_A \cos(nxy) \, d\lambda_2=0$

I need to show that $$\lim_{n\rightarrow \infty} \int\limits_A \cos(nxy) \, d\lambda_2=0$$ for every Borel set $A\subset \mathbb{R}^2$ which has finite Lebesgue measure. I tried to use the definition ...
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2answers
33 views

Prove that $\mathcal M : = \left \{E \in \mathcal B_{\Bbb R^2}\ |\ \lambda_{\Bbb R^2} (E+x) = \lambda_{\Bbb R^2} (E) \right \}$ is a monotone class.

Let $(\Bbb R^2, \mathcal L_{\Bbb R^2}, \lambda_{\Bbb R^2})$ be the Lebesgue measure space on $\Bbb R^2.$ Prove that $$\mathcal M : = \left \{E \in \mathcal B_{\Bbb R^2}\ |\ \lambda_{\Bbb R^2} (E+x) = ...
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1answer
20 views

Show that $f$ is measurable with respect to $L$.

Let $(R,L, λ)$ be the Lebesgue measure space and $f : R → R$ a function with $f(x) = 127e^x$ for all $x ∈ Q^c$. Show that $f$ is measurable with respect to $L$. This is a question on a past exam paper ...
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2answers
48 views

prove that $f_n\in L^1_\text{loc}(\mathbb{R})$

Today I have a exercise about locally integral as following : Defined ($n\in\mathbb{N^{*}}$ : $$f_{n}(x)=\begin{cases}\log |x| , &|x|>\frac{1}{n} \\ -\log n , &|x|<\frac{1}{n}\end{cases}$...
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0answers
35 views

Lebesgue measure of a subset [$0,1$]

Determine the Lebesgue measure of the subset of [$0,1$] whose elements have neither first nor second digit in their decimal expansion equal to "$0$" My workings for this problem are as ...
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1answer
27 views

Prove that $f$ is Lebesgue integrable on $E$ if and only if $\sum_{n=0}^\infty 2^nm(\{x\in E:f(x)\geq2^n\})<\infty$

Question: Let $E$ be a finite measure space and let $f$ be a nonnegative function on $E$. Prove that $f$ is Lebesgue integrable on $E$ if and only if $\sum_{n=0}^\infty 2^nm(\{x\in E:f(x)\geq2^n\})&...
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chapter 2 problem B 27: frank jones lebesgue integration on euclidean spaces

prove that $$\lambda^{*}(A) = \inf \Big\{\sum_{k=1}^{\infty} \lambda(I_k) \Big| A \subset \bigcup_{k=1}^{\infty} I_k \Big\}$$ It suffices to show that $$\forall \epsilon > 0 , \exists (I_k) \text{ ...
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2answers
28 views

Determine liminf$A_{n}$ and limsup$A_{n}$ for the $A_{n}$ of the following sets

$$A_{n} = \begin{cases} (\frac{1}{n}-2 , 1) \ \ \ \ \ \text{if} \ n \ \text{is odd} \\ (0,3 + \frac{1}{n}) \ \ \text{if} \ n \ \text{is even} \end{cases}$$ This question is on a past measure theory ...
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1answer
15 views

Convergence of integrals of function sequences evaluated by inequalities. [closed]

If the function sequence $\{f_n\}_{n=1}^{\infty},\{g_n\}_{n=1}^{\infty},\{h_n\}_{n=1}^{\infty}$ on $L^1(\mathbb{R})$ satisfies $$f_n\rightarrow f,\ g_n\rightarrow g,\ h_n\rightarrow h\ \text{a.e.}\ \...
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1answer
61 views

Approximation of open subsets of $\Bbb R^2$ by compact sets.

I am having difficulty to prove one of the regularity properties of Lebesgue measure. Here it is $:$ Let $(\Bbb R^2, \mathcal L_{\Bbb R^2}, \lambda_{\Bbb R^2})$ be the Lebesgue measure space on $\Bbb ...
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1answer
49 views

Show that $\mathcal A_1$ $\cap$ $\mathcal A_2$ is also a $\sigma$-algebra

I am currently studying for a measure theory final and have come across a past short exam question that reads "Let $\mathcal A_1$ and $\mathcal A_2$ be a $\sigma$-algebra of subsets of a set X. ...
6
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1answer
95 views

Integrable function $f$ such that $\int_I f(x)dx=0$ for intervals of arbitrarily small length.

A past qual question from my university reads: Let $f$ be an integrable function satisfying $\int_0^1 f(x)dx=0$. Prove that there are intervals $I$ of arbitrarily small positive length such that $$\...
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1answer
22 views

Let X be a set. How many $\sigma$-algebras of subsets of X contain exactly $5$ elements?

One of the questions on a past final for a measure theory course I'm taking is "Let X be a set. How many $\sigma$-algebras of subsets of X contain exactly $5$ elements?". Would the answer to ...
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1answer
56 views

List all the possible values for $\int_{\mathbb{R}}\sup_{k\in\mathbb{N}}f_k(x)dx$ under these conditions…

Question: Let $\{f_k(x)\}_{n=1}^\infty$ be a sequence of nonnegative functions on $\mathbb{R}$ such that $\sup_{x\in\mathbb{R}}f_k(x)=\frac{1}{k}$, and $\int_{\mathbb{R}}f_k(x)dx=1$. List all the ...
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1answer
45 views

Property of a positive Lebesgue measure set in $\mathbb{R}^2$

Let $A\subset \mathbb{R}^2$ be a closed set of positive Lebesgue measure. Can we find positive Lebesgue measure sets $A_1,A_2\subset \mathbb{R}$ such that $A_1\times A_2\subseteq A$? Note that the ...
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1answer
43 views

Does it make sense to define the length of a line segment in terms of addition of infinite points?

Since a point has zero length, how can a line segment of, say, 1-unit length—which is a collection (addition) of infinite points, that is $0 + 0 + \cdots$—have 1-unit length? Does it make sense to say ...
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2answers
28 views

A question about finding Lebesgue measure of a specific set I am unable to find

This is a quiz question of previous year asked in my measure theory exam and I am unable to solve it. Let $k$ be a positive integer and let $$S_{k} = \{x \in [0, 1] | \text{ a decimal expansion of $x$...
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0answers
37 views

Lebesgue measure and Left and right inverse distribution functions

Let $\lambda$ be the Lebesgue measure and $F(x)$ be a distribution function (i.e. right continuous and non decreasing with range [0,1]. Show that $$ \lambda\big\{u\in (0,1]: F_l^{\leftarrow}(u) \neq ...
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0answers
18 views

Evaluate the induced measure $mT^{-1}(A)$ of the set $A$ in the following cases.

Let $m(\cdot)$ denote the Lebesgue measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $T:\mathbb{R}\to\mathbb{R}$ be the map $T(x) = x^{2}$. Evaluate the induced measure $mT^{-1}(A)$ of the set $A$...
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1answer
21 views

Show $0<\epsilon<1$ there exists some $\delta_{p,\epsilon}>0$ such that $m(\{x\in X:|f(x)|>\epsilon\})\geq\delta_{p,\epsilon}$ for each $f\in E_p$.

Question: Let $(X,A,m)$ be a measure space such that $m(X)=1$. For each $1<p<\infty$ define the set $E_p=\{f\in L^1(m):\int |f|dm=1 \text{ and} \int |f|^pdm=2\}$. Show that for each $0<\...
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2answers
98 views

Given $f$ is a Lebesgue measurable function and $\int_0^1 x^{2n}f = 0 ~~~ \forall n$ , then show that $f = 0$ a.e.

Given $f$ is a Lebesgue measurable function and $\int_0^1 x^{2n}f\,d\mu = 0 \quad \forall n$, then show that $f = 0$ a.e. Of course, if it was given that $f \geq 0$ then this was pretty trivial. My ...
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1answer
45 views

Prove that $\lim_{j\rightarrow\infty}\int_1^\infty\frac{f_j(x)}{x}dx=\int_1^\infty\frac{f(x)}{x}dx$ under these conditions…

Question: Let $\{f_j\}_{j\in\mathbb{N}}$ be a sequence of Lebesgue measurable functions satisfying $$\sup_{j\in\mathbb{N}}\int_1^\infty f_j^2(x)dx\leq1$$ such that $f_j\rightarrow f$ pointwise a.e. ...
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1answer
36 views

Is it necessarily true that $\int_0^1\frac{1}{(x+f_n(x))\log(n+1)}dx\rightarrow0$ as $n\rightarrow\infty$ under these conditions

Question: Assume the sequence of Borel measurable functions $f_n:(0,1)\rightarrow\mathbb{R}$ satisfies $f_n(x)\geq\frac{1}{n}$ and $f_n(x)\rightarrow\infty$ as $n\rightarrow\infty$ for every $x\in(0,...
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1answer
47 views

Integral non-zero only if integrand is to the power of two

Is it possible to find a probability space $(\mathbb{C}, \Sigma, \mathbb{P})$, where $\mathbb{C}$ is the complex plane and $\Sigma$ is the Borel sigma algebra, and a measurable function $F$ such that,...
2
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0answers
42 views

Lebesgue measure on $\mathbb{R}^2$

Let $P=A_1\times A_2,$ where $A_1,A_2\subset \mathbb{R}$ are set of positive Lebesgue measure, and $Z\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue ...
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1answer
39 views

Show that $\int_0^1\frac{1}{|f(x)-x_0|}dx$ is unbounded

Let $f$ be a Lebesgue measurable functinon on such that $f:[0,1]\rightarrow [0,1]$. Prove that for any $M$ there exists $x_0\in[0,1]$ such that $$ \int_0^1\frac{1}{|f(x)-x_0|}dx\geq M. $$ My attempt: ...
2
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2answers
26 views

$\rho(f,g)=\int_E \min(1,|f-g|)dm$. Prove that $f_n$ converges to $f$ in measure if and only if $\rho(f_n,f)\rightarrow 0$ as $n\rightarrow\infty$

Question: Suppose $m$ is a finitemeasure on a measurable space $E$. Define $\rho(f,g)=\int_E \min(1,|f-g|)dm$. Prove that $f_n$ converges to $f$ in measure if and only if $\rho(f_n,f)\rightarrow 0$ ...
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2answers
55 views

What is the Lebesgue measure of the Koch and the Minkowski curves?

Are the Koch curve and the Minkowski curve Lebesgue measurable? (I believe they are.) If so, what are their measures? (Intuitively, it would seem to be zero.) I unfortunately can't seem to find much ...
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0answers
16 views

Do all metric spaces have to be measurable sets or is there way to have a non-measurable metric space?

Since it is called a "distance-function" and distances are "measured", I wonder if the definitions match the language.
2
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0answers
27 views

Help with finding the Lebesgue decomposition of measures

Consider the increasing, right-continuous function $$ F(x) = \begin{cases} 0 &x < 0 \\ 1+x &x \geq 0 \end{cases} $$ and let $\nu = \nu_F$ be the associated Borel ...
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1answer
32 views

Is this a reasonable measure for comparing the sizes of two sets of real numbers?

Introduction We want to define a measure for determining whether one subset of $\mathbb{R}$ is bigger than the other. I am have defined such a measure, but I am not sure if it is a very useful one. I ...
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1answer
54 views

A few questions regarding using Markov/Chebyshev and writing the Lebesgue measure in terms of the Lebesgue Integral

I just asked a question here Prove that $m(\{x\in[0,1]:\lim \sup_{j\rightarrow\infty}f_j(x)\geq\frac{1}{2}\})\geq\frac{1}{2}$ under these conditions..., and I am having a hard time seeing the bounds ...
2
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2answers
51 views

Prove that $m(\{x\in[0,1]:\lim \sup_{j\rightarrow\infty}f_j(x)\geq\frac{1}{2}\})\geq\frac{1}{2}$ under these conditions…

Question: Suppose for each $j\in\mathbb{N}, f_j:[0,1]\rightarrow\mathbb{R}$ is Lebesgue measurable such that $0\leq f_j\leq\frac{3}{2}$ and $\int_0^1 f_j dm=1$. Prove that $m(\{x\in[0,1]:\lim \sup_{j\...
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1answer
22 views

An exponential family of probability distributions have densities that are defined relative to a measure. What is a measure in this context?

I would like to understand the nature of measure, using the exponential family of probability distributions as a context (because I understand the latter well). I understand that we want equation (8.1)...
4
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1answer
39 views

To show some set is positive Lebesgue measure

Let $P$ be a set of positive Lebesgue measure in $\mathbb{R}^n$ and $O$ be an open set in $\mathbb{R}^n$ such that $E=P\cap O$ is a set of zero Lebesgue measure. Can we conclude that $P\setminus \...
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0answers
19 views

Existence of a self-similar symmetric subset of the unit square with non-trivial Lebesgue measure

I am trying to find out whether there exists a Borel set $F\subset[0,1]^2$ that is strictly self-similar and symmetric and such that $0<\lambda(F)<1$, where $\lambda$ is the Lebesgue measure. We ...
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1answer
76 views

Property of Lebesgue measure in $\mathbb{R}^n$

Let $x\in \mathbb{R}^n$ and $A\subset \mathbb{R}^n$ be a set of positive Lebesgue measure and for any $r>0$ the set $$B(x,r)\cap A$$ is non empty. Let $D^c$ be a measure zero set and hence $D$ is ...
2
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0answers
45 views

$f:\mathbb{R}\rightarrow\mathbb{R^2}$ such that $|f(x)-f(y)|\leq\sqrt{|x-y|}$ for every $x,y\in\mathbb{R}$. Show $m_2(f(A))=0$

Question: Suppose $A\subset\mathbb{R}$ such that $m_1(A)=0$, where $m_1$ is the one-dimensional Lebesgue measure. Now, consider a function $f:\mathbb{R}\rightarrow\mathbb{R^2}$ such that $|f(x)-f(y)|\...

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