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).

17
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1answer
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Construct a Borel set on R such that it intersect every open interval with non-zero non-“full” measure

This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis. The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and ...
2
votes
3answers
523 views

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
45
votes
2answers
12k views

Lebesgue measurable but not Borel measurable

I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
8
votes
2answers
378 views

measurable functions and existence decreasing function

Let $f(t)$ be a measurable and almost everywhere finite function, defined on the closed interval $E = [a, b]$. Prove the existence of a decreasing function $g (t)$, defined on [a, b], which satisfies ...
9
votes
3answers
4k views

Lebesgue density theorem in the line

Suppose $A \subseteq \mathbb{R} $, $m(A) > 0 $. Then for almost all $x \in A $ we have $$ \lim_{\epsilon \to 0^+ } \frac{ m(A \cap (x - \epsilon, x + \epsilon))}{2 \epsilon} = 1.$$ Can someone ...
14
votes
4answers
3k views

Prove Borel Measurable Set A with the following property has measure 0.

This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition. Let $\epsilon \in (0,1)$, let $m$ be Lebesgue measure, and suppose $A$ is Borel Measurable subset ...
19
votes
1answer
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Vitali set of outer-measure exactly $1$.

I know that for any $\varepsilon\in (0,1]$ we can find a non-measurable subset (w.r.t Lebesgue measure) of $[0,1]$ so that its outer-measure equals exactly $\varepsilon$. It is done basicly with the ...
32
votes
5answers
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What is wrong in this proof: That $\mathbb{R}$ has measure zero

Consider $\mathbb{Q}$ which is countable, we may enumerate $\mathbb{Q}=\{q_1, q_2, \dots\}$. For each rational number $q_k$, cover it by an open interval $I_k$ centered at $q_k$ with radius $\epsilon/...
1
vote
1answer
200 views

Prove $X(\omega) = \sup\{y \in \mathbb{R}: F(y) < \omega\}$ is a random variable.

Let F be a distribution function. On $(\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ denotes Lebesgue measure. Define X: $\Omega \to \mathbb{R}$ by $X(\omega) = \sup(y \...
18
votes
4answers
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Prove that every Lebesgue measurable function is equal almost everywhere to a Borel measurable function

Suppose $(\mathbb{R},\Sigma(m),m)$ is our measure space, where $m$ is Lebesgue measure. Also, suppose $f : \mathbb{R} \to [-\infty, \infty]$ is a Lebesgue measurable function. The problem: Prove ...
9
votes
5answers
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Finite additivity in outer measure

Let $\{E_i\}_{i=1}^n$ be finitely many disjoint sets of real numbers (not necessarily Lebesgue measurable) and $E$ be the union of all these sets. Is it always true that $$ m^\star (E)=\sum_{i=1}^N m^...
11
votes
2answers
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Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable ...
3
votes
2answers
314 views

$E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$

Ran across this problem and need some help. Let $E$ be a measurable subset of the real numbers and suppose that for any open interval $I$ one has $m(E\cap I)\le \frac{1}{2}m(I)$, where $m$ is the ...
5
votes
1answer
363 views

Set $E\subset \mathbb{R}^n$ of positive Lebesgue measure such that the Lebesgue measure of its boundary is zero

Let $E\subset \mathbb{R}^n$ have positive Lebesgue measure. What are easily interpretable sufficient conditions on $E$ to guarantee that the difference between the closure $\bar{E}$ and the interior $\...
5
votes
2answers
4k views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and $\...
6
votes
2answers
1k views

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

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise almost everywhere convergent subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ? Edit: I am adding my ...
6
votes
3answers
1k views

How can I find a subset of a set with “half the size” of the original?

let $E\subset\Bbb R$ Lebesgue-measurable with $m(E)\lt \infty $, prove that exist $A\subset E$ Lebesgue-measurable such that $m(A)=m(E)/2$. I've tried this For every $k\ge 1$ exist $\{I_n^k\}$ with ...
1
vote
1answer
357 views

Let $g:\mathbb{R}\to\mathbb{R}$ be a measurable function such that $g(x+y) =g(x)+g(y).$ Then $g(x) = g(1)x$ . [closed]

Let $g:\mathbb{R}\to\mathbb{R}$ be a measurable function such that $$g(x+y) =g(x)+g(y).$$ How to prove that $g(x) = cx$ for some $c\in \mathbb{R}?$ The main thing to do here relies upon the fact ...
2
votes
1answer
302 views

Proof of integral of a simple measurable function

I need to prove that for $x_{1}, x_{2}, \cdots $ and disjoint $A_{1}, A_{2}, \cdots $, for a simple measurable function $f(\omega) = \sum_{i=1}^{\infty}x_{i}I_{A_{i}}(\omega)$, $$ \int f d \mu = \int_{...
0
votes
2answers
374 views

$E$ Lebesgue measurable implies $E^2$ Lebesgue measurable?

Suppose $E \subset \mathbb{R}$ is Lebesgue measurable. Define $$ E^2 = \{x^2 : x \in E\}. $$ Is $E^2$ Lebesgue measurable as well? I believe the answer is yes, but I am struggling to prove it. I ...
8
votes
2answers
7k views

Example of a general random variable with finite mean but infinite variance

Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, ...
8
votes
4answers
952 views

Why do we consider Borel sets instead of measurable sets?

Dumb/Challenging conventional wisdom question possibly related to my previous question. Why do we sometimes consider a measure space $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{B}(\mathbb{R}), \lambda)$...
7
votes
3answers
1k views

The Lebesgue measure of zero set of a polynomial function is zero

Suppose $f :\mathbb R^n \to \mathbb R$ be a non zero polynomial(more generally smooth) function.Suppose $Z(f)=\{ x \in \mathbb R^n \mid f(x)=0 \}$.Show that Lebesgue measure of $Z(f)$ is zero. I am ...
6
votes
1answer
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Prove the Countable additivity of Lebesgue Integral.

Let $E\subset\mathbb{R}$ a measurable subset, $f\in L^1(E)$ and $\{E_n\}$ a disjoint countable union of measurables sets such that $\bigcup E_n=E$. Show that $$ \int_Ef=\sum_{n=1}^\infty\int_{E_n} f$$ ...
6
votes
2answers
821 views

If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

Let $\left(X,\tau\right)$ a topology space and $\mathcal{B}$ a base of the topology, my question is: The Borel $\sigma$-algebra is generated by $\mathcal{B}$ ?
1
vote
2answers
395 views

Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$

Reference:(Fubini's Theorem) Question: Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$ ? Edit: Both ...
6
votes
1answer
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Let $f$ be a bounded measurable function on $E$. Show that there are sequences of simple functions converging uniformly on $E$.

Let $f$ be a bounded measurable function on $E$. Show that there are sequences of simple functions on $E$, $\{\phi_n\}$ and $\{\psi_n\}$, such that $\{\phi_n\}$ is increasing and $\{\psi_n\}$ is ...
5
votes
1answer
458 views

Outer measure of a nested sequence of non-measurable sets

Let $\bigcup_{n=1}^\infty E_n=E$ and $ E_{n} \subseteq E_{n+1} $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E) $ even if each $E_n$ is a non-measurable set, where $\mu^*$ is Lebesgue ...
4
votes
1answer
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Show that a function almost everywhere continuous is measurable

I want to prove that a function $f:\mathbb{R}^n\rightarrow \overline{\mathbb{R}}$ that is continuous everywhere except for a set $E$ of Lebesgue measure zero is a Lebesgue measurable function. We know ...
2
votes
1answer
405 views

Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0<\alpha<1$ an interval $I$ such that $m(A\cap I)/m(I)>\alpha$.

Let $A\subset\mathbb{R}$ a measurable where $0<m(A)<\infty$. Show that exists for each $0<\alpha<1$ an interval $I$ such that $$ \frac{m(A\cap I)}{m(I)}>\alpha. $$ MY ATTEMPT: ...
2
votes
1answer
2k views

Cartesian Product of Borel Sets is Borel Again

Let $E$ and $F$ be Borel measurable subsets of $\mathbb R^{d_1}$ and $\mathbb R^{d_2}$, respectively. Then $E \times F$ is also Borel measurable in $\mathbb R^{d_1 + d_2}$. I suppose it is necessary ...
1
vote
1answer
7k views

f a real, continuous function, is it measurable?

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function. I need to show that is a measurable function. I tried working with the definition: Let $f: X \to \mathbb{R}$ be a function. If $f^{-1}(...
7
votes
1answer
1k views

Measure of image of critical points set is equal 0

Let $f:\mathbb{R}\to \mathbb{R}$ be $C^1$ function and $K = \{x : f'(x) = 0 \} $. Show that $\mu \left(f\left(K\right)\right) = 0$, where $\mu$ is Lebesgue measure. My attempt was following: $$\mu \...
5
votes
2answers
456 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for $...
2
votes
1answer
222 views

Lebesgue-measurable sets requiring the Axiom of Choice to construct

Every known construction of the Vitali set relies on the Axiom of Choice. It happens to not be Lebesgue-measurable. Must every set whose construction relies on the Axiom of Choice not be Lebesgue-...
1
vote
1answer
291 views

Let $\lambda(A)$ be the Lebesgue measure of $A$. There exists an interval $I$ such that $\lambda(E \cap I) / \lambda(I) < 1-\epsilon$

(Not mentioned in title but $\epsilon$ is a number greater than $0$ and $E$ a Lebesgue measurable subset of $\mathbb{R}^n$.) I know a question equivalent to this one has been asked (here). But it was ...
0
votes
2answers
212 views

Is $\delta$ in $L^\infty$?

I think the question title says is all. I am wondering, is the Dirac delta in the Lebesgue space $L^\infty$?
0
votes
1answer
1k views

Example of decreasing sequence of sets with first set having infinite measure

I was wondering if someone could please give me an example of a sequence of decreasing sets where the first set has infinite Lebesgue measure; i.e., $\{B_{n}\}_{n=1}^{\infty}$ such that $m(B_{1}) = \...
29
votes
3answers
13k views

What is the difference between outer measure and Lebesgue measure?

What is the difference between outer measure and Lebesgue measure? We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\...
20
votes
7answers
599 views

When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail

We know that if $f\in L^1(\mathbb{R})$, then $\|f(\cdot+1/n)-f(\cdot)\|_{L^1}\to 0$ as $n\to \infty$, which implies that there exists a subsequence $f_{n_k}=f(x+\frac{1}{n_k})$ such that $f_{n_k}\to f$...
23
votes
2answers
6k views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ be a ...
10
votes
1answer
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A rigorous meaning of “induced measure”?

In my readings I often come across terms like "induced measure" or "induced Lebesgue measure". For example: $$\int_{\mathbb{B}^n}u\frac{\partial v}{\partial x_j}\;dx = \int_{\mathbb{S}^{n-1}}uv\...
7
votes
2answers
3k views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
8
votes
1answer
4k views

Non measurable subset of a positive measure set

I am self-studying measure theory and I have seen this theorem: If $A$ is a set of positive measure, then there exists a subset $D$ of $A$ that is non measurable. I am not sure how to prove it. I ...
1
vote
1answer
984 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
12
votes
2answers
3k views

Nowhere dense subsets of $[0,1]$ with positive measure other than fat Cantor sets

This is my first time on the board, so forgive me if I've posted incorrectly. In any case, I think my title is self-explanatory: the only examples I've encountered for nowhere dense subsets of $[0,1]$ ...
11
votes
3answers
6k 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 ...
8
votes
3answers
1k views

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 ...
7
votes
1answer
136 views

What is the Sigma Algebra generated by Jordan measurable sets?

Unlike Lebesgue measurable sets, Jordan measurable sets do not form a Sigma algebra. So my question is, what is the Sigma algebra $J$ generated by Jordan measurable sets? All intervals are Jordan ...
6
votes
1answer
520 views

Converse for Fubini-Tonelli's theorem

By Fubini-Tonelli's theorem, we know that if $E\subset \mathbb{R^{n+m}}$ and $f: \mathbb{R^{n+m}}\to \mathbb{R_{>0}}$ are measurable and $f$ integrable, then the sections $E_x=\{y\in \mathbb{R^m}: (...