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).

learn more… | top users | synonyms

4
votes
3answers
32 views

Show that if $A,B$ are measurable, $A\subset E\subset B$, and $m(A)=m(B)$, then $E$ is measurable.

Here's the full problem: Suppose $A\subset E\subset B$, where $A,B$ are measurable with finite measure. Show that if $m(A)=m(B)$, then $E$ is measurable. Here, we are dealing with measure space ...
2
votes
0answers
21 views

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k ...
0
votes
1answer
24 views

Measure of the graph of a function such that the graph does not have measure zero.

In an exercise for class we were asked to prove that the graph of a continuous measurable function has measure zero. Ok, so let us just look at some measurable function that is not necessarily ...
0
votes
1answer
44 views

Show that the measure is zero

I am asked to show that the $2-$dimensional Lebesgue measure of the graph of a continuous real function is zero. Could you give me some hints how I could show it??
2
votes
0answers
16 views

Measure Products Probability

Let $X$ and $Y$ be independent random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $F_X$ and $F_Y$ be their distribution functions and $\mu_X$ and $\mu_Y$ their laws. ...
0
votes
1answer
13 views

Approximate measurable function by simple function with compact support

Let $f$ be a nonnegative Lebesgue measure function on $\mathbb{R}$, $\epsilon>0$. How can we approximate $f$ by a nonnegative simple function $s$ with compact support s.t. $s\leq f$ and ...
0
votes
0answers
31 views

Characterization of Lebesgue measure based on translation invariant

I am trying to solve a problem about characterization of Lebesgue measure. Let $(\mathbb{R}^n, \mathcal{B}, \mu)$ be a Borel measure space whose measure $u$ is translation invariant and exist a set ...
0
votes
1answer
18 views

Measure of sum of sets of “Cauchy” sequence bounded?

Let $\{A_n\}_n$ be a sequence of sets of a $\delta$-ring $\mathfrak{M}$ of measurable sets with finite Lebesgue measure. Let us suppose that $$\forall\varepsilon>0\quad\exists ...
0
votes
1answer
39 views

What does it mean m(dx), where m is Lebesgue measure?

Let a $\in \mathbb{R}$, $\phi_n : \mathbb{R} \rightarrow \mathbb{R}_+$, $\phi_n (x) : = \frac{n}{\sqrt{2 \pi}} e^{\frac{-n^2 x^2}{2}}$, $n \geq 1$ and let $\mu_n (d x) : = \phi_n (x - a) \lambda (d ...
1
vote
0answers
39 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
2
votes
2answers
72 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
1
vote
1answer
35 views

Lebesgue integral and anti-derivative

For which Lebesgue measures the Lebesgue integral of a differentiable function over a Euclidean space or an orientable manifold coincides with its anti-derivative? For example, can we find the class ...
0
votes
2answers
38 views

Why is the outer measure of the set of irrational numbers in the interval [0,1] equal to 1?

Just learned Lebesgue outer measure from Royden's Real Analysis. Let me give my proof. First, let $A$ be the set of irrational numbers in [0,1]. So $A\subset [0,1]\Rightarrow m^*(A)\le ...
1
vote
0answers
54 views

Is there a dense set of positive measure which does not contain any open set?

I want to construct $A\subseteq[0,1]$ with $m(A)>0$ and for every open subset $U$ of $[0,1]$, $0<m(A\cap U)<m(U)$ and $U\not\subseteq A$. I think these sets must have measure zero. ...
1
vote
1answer
30 views

Prove $f^{-1}(B)\in\mathcal{A}$ where Borel set $B$, $\sigma$-algebra $\mathcal{A}$, $\mathcal{A}$-measurable function $f$.

I am trying to solve a homework problem in "Lebesgue integration" course. And I found the theorem seems to be useful to my problem which is : Let $\mathcal{A}$ be a $\sigma$-algebra of ...
1
vote
1answer
64 views

Show that the measure is Lebesque

I want to show that a measure is the same as the Lebesque measure. How can I do that?? What properties does this measure has to satisfy so that it is Lebesque??
1
vote
2answers
37 views

Show that the measure is equal to zero

Let $\mu$ be a Borel measure in $\mathbb{R}$ such that $\mu(I) \leq v^a(I)$ for each bounded interval $I$, where $a>1$. Show that $\mu=0$. ($v(R)$ is the volume of $R$) Do we maybe use the ...
0
votes
0answers
20 views

Existence of a A measurable function

Let A be sigma algebra having subsets of R only. We define a function from subset of A to R is said to be A measurable iff every borel set is pulled back to elements of A. Is there a sigma algebra B ...
2
votes
1answer
25 views

question on existence of open set

Let $U$ be a bounded open set in $\mathbb{R}^n$ and $A$ be an open subset of $U$. Fixed $\epsilon >0$. Does there exist an open set $B \subset U$ such that $B \cap \overline{U} \ne \emptyset$ and ...
3
votes
0answers
36 views

Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
1
vote
3answers
37 views

ONB: Fourier Series

Given the Hilbert space $L^2([-\pi,\pi])$. Consider the orthonormal system: $$\mathcal{S}:=\{\frac{1}{\sqrt{2\pi}}e^{ikx}:k\in\mathbb{Z}\}$$ This is an ONB. How do I prove this? I guess, I could try ...
2
votes
1answer
51 views

A question on Lebesgue measure: Inequalities

Let $A_n$ be a decreasing sequence of Borel sets with finite but with measure $\geq \epsilon > 0$ for all $n$. Then there exists a compact set $B_n \subset A_n$ for all $n$ such that $\mu(A_n ...
0
votes
1answer
29 views

Is $\mathbb{R}$ is the union of a negligible and a meager set?

A subset of $\mathbb{R}$ is said to be "negligible" iff it is of Lebesgue measure zero, and "meager" iff it is contained in a countable union of nowhere dense closed sets (i.e., closed sets with empty ...
1
vote
0answers
24 views

Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
0
votes
0answers
28 views

Integrability of a function from Stein Shakarchi Real Analysis

I have a question from Stein+ Shakarchi's Real Analysis book regarding the integrability of this particular function. (pg. 63-64) Consider the function $$f(x)= \begin{cases} \frac{1}{ \mid x \mid ...
4
votes
2answers
66 views

How to calculate $\lim_{n\rightarrow \infty } \int ^{\infty}_{0}\frac{n\sin \left(\frac {x}{n}\right)}{\left(1+\frac {x}{n}\right)^{n}}dx$

The original is to calculate $$\lim_{n\rightarrow \infty } \int ^{\infty}_{0}\dfrac{n\sin \left(\frac {x}{n}\right)}{\left(1+\frac {x}{n}\right)^{n}}dx$$ or give a integral form. I guess Lebesgue's ...
4
votes
1answer
159 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
8
votes
1answer
98 views

Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} ...
3
votes
1answer
48 views

Counterexample to "if $\int_E f < \infty$, then $\lim_{n \to \infty} \int_A f_n = \int_A f$

Part a) of the question is as follows: "Suppose that $E \subset \mathbb{R}^d$ is a measurable set and that $f, f_n$ are measurable functions on $E$ satisfying $f_n \to f$ a. e. on $E$. Suppose that ...
2
votes
0answers
17 views

Set of zeros of derivate - Lebesgue measure [duplicate]

I'm currently struggeling with the following: Let $\lambda$ be the Lebesgue measure and $f \in C^2[0, 1]$. Show: If $\lambda(\{x \in [0, 1]; f(x) = 0 \}) > 0$, then $\lambda(\{ x \in [0, 1]; f'(x) ...
5
votes
1answer
48 views

Prove that this sequence converges almost surely

Suppose that $(X_n)_{n\ge1}$ is a sequence of independent random variables with $E[|X_n|] < \infty$ for all $n$ and $E(X_n) = \mu$. Prove that $$\sum_{n=1}^{\infty}\frac{1}{2^n}X_n = \mu \; a.s$$ ...
1
vote
1answer
10 views

Modification of Set Function in Construction of Lebesgue Measure

Suppose in the construction of Lebesgue measure we replace the set function $\mu((a,b))=b-a$ with $\mu((a,b))=\sqrt{b-a}$. What can we say about $\mu^*$ and the $\sigma$-algebra of measurable sets? ...
2
votes
1answer
41 views

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 ...
0
votes
1answer
21 views

Is Borel measurable function composed with Lebesgue measurable function a Lebesgue measurable function?

Let $g$ be a Borel measurable function and $f$ be a Lebesgue measurable function. Then, is $g(f(x))$ a Lebesgue measurable function?
2
votes
0answers
25 views

Outer Measure in Cantor-like Set

Consider the Cantor-like set $C$ resulting from the Cantor-like construction, which starts with $k$ disjoint closed intervals $\delta_i$, $k \ge 2$, $i=1,\dots,k$ of the unit interval. Given an ...
3
votes
1answer
32 views

Do positive integrals imply positive function in this case?

Suppose that $f: \mathbb{R} \to [0, \infty)$ is Borel measurable and satisfies $\|f\|_\infty \le 1$ and $\|f\|_1 = 1$. If $$\int_a^b \! f(x) \, dx > 0$$ for all $a < b$, does it necessarily ...
1
vote
1answer
37 views

Assumptions of the MCT

Monotone Convergence Theorem: Let ($f_n$) be a sequence in $\Sigma^+$ (i.e. measurable and nonnegative), such that $f_{n+1}\geq f_n$ almost everywhere for each $n$. Let $f=\limsup_n f_n$. Then $\mu ...
1
vote
2answers
34 views

Outer Measure Subadditivity

I'm having trouble constructing a sequence $\{E_n\}$ of disjoint subsets of $\mathbb{R}$ such that $$m^{*}\left(\bigcup_{i}E_i\right) < \sum_{i}m^{*}(E_i).$$ What's a way to gain some intuition ...
3
votes
0answers
57 views

Can $\|f\|_p\to\infty$ arbitrarily slowly? (Looking for hints.)

Given $f$ is Lebesgue measurable on $(0,1)$ and not essentially bounded, is it true that to every positive function $\Phi$ on $(0,\infty)$ such that $\Phi(p)\to\infty$ as $p\to\infty$ one can find an ...
3
votes
3answers
48 views

Does a set of positive outer measure contain a measurable subset of positive measure?

Is it true or false that whenever $E \subseteq \mathbb{R}$ is such that $m^*(E) > 0$, where $$ m^*(E) = \inf\left\{\sum_{n=1}^\infty|b_n - a_n|\, :\mid\, E \subseteq \bigcup_{n = ...
0
votes
1answer
35 views

Determine $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$, and is $u$ $\mathcal B(\mathbb{R})/\mathcal B(\mathbb{R})$-measurable?

Let $u:\mathbb{R}\to\mathbb{R}$ be given by $u(x)=\left \lfloor x \right \rfloor$. Determine the set $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$. Show that $u$ is $\mathcal ...
1
vote
1answer
37 views

How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
1
vote
1answer
55 views

Lesbegue Outer Measure

Consider the unit interval $I=[0,1]$ and let $\mathcal{M}$ be the $\sigma$-algebra of all Lebesgue measurable subsets of $I$. Denote by $m_*$ the Lebesgue outer measure on $\mathcal{M}$. Suppose that ...
2
votes
2answers
42 views

Decomposing Countable Union of Measurable Sets

Why can every set $E$ in the real numbers with $\mu^{*}(E)=\infty$ be realized as the disjoint union of countably many measurable sets, each of which has finite outer measure? I'm trying to see this ...
1
vote
1answer
19 views

Algebra Generated by Open and Closed Intervals

If $E$ is the collection of all open intervals $(a,b)$ in $X=[0,1]$, how do I know that the $\sigma(E)$ contains all closed intervals $[a,b] \subset X$, in particular closed intervals involving the ...
3
votes
1answer
95 views

Union of uncountable measurable sets is bounded?

I'm trying to prove that if $ \{A_r\; :\: r>0\}$ is a family of measurable subsets of $\mathbb{R^n}$ such that $A_r \subset A_s$, if $r<s$, then uncountable union $\cup_{r>0} A_r$ is ...
1
vote
2answers
40 views

Determine whether the function is Lebesgue measurable

Here's the full problem: Let $\mathcal{N} \subset [0,1]$ be a non-measurable set. Determine whether the function $$f(x) = \left\{ \begin{array}{lr} -x & : x \in \mathcal{N}\\ ...
1
vote
1answer
12 views

Help understanding the proof of $\mu$-completion of a sigma algebra $\mathfrak M$

The theorem (from Rudin) states: For a measure space $(X,\mathfrak M, \mu)$, let $\mathfrak M^*$ be the collection of all $E\subset X$ for which there exists sets $A$ and $B\in\mathfrak M$ such that ...
0
votes
1answer
29 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
0
votes
1answer
27 views

Show that $\{\hat {f _n }\} $ converges in measure to $f$

I want to show that if $f_n $ converges in measure to $f $ and for each $n$ $\mu ( \{x : |\hat {f _n (x)} -f_n(x) \ge \frac {1 } {n } \} )=\frac {1 } {n } $, then $\{\hat {f _n }\} $ converges in ...