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
12 views

Piecewise-$C^k$ function

A function $f:\mathbb{R}_+\to \mathbb{R}^n$ is called piecewise-$\mathcal{C}^k$ if There is a partition $\{ x_i \}_{i=0}^\infty $ with $x_0=0$ and $x_{i+1}>x_i$ such that $f$ is $\mathcal{C}^k$ ...
1
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0answers
28 views

Fubini's theorem for a measure on product space, which is not a product of measures

Let $X,Y$ be some nice measurable spaces (I'm interested in $[0,1]$ so we can assume compact, etc.). Let $\mu$ be a measure on $X\times Y$ (again, assume it's a nice probability measure, or even ...
3
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0answers
43 views

Is a measure on product space necessarily a product of measures?

Let $X,Y$ be some nice measureable spaces (i'm interested in $[0,1]$ so we can assume compact, etc.). let $\mu$ be a measure on $X\times Y$.(again, assume it's nice, i.e. probability measure. anything ...
9
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1answer
230 views

What is the cardinal of non measureable set?

we know that $|P( \Bbb{R} )|=|L (\Bbb{R} )|$ ( $L (\Bbb{R} ) $ is the set of all Lebesgue measureable set ) note that $L (\Bbb{R} ) \subsetneq P( \Bbb{R} ) $. What is the cardinal of non ...
0
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2answers
90 views

Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$ \mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?. $$ No ideas how to start this one. I see that the limit of ...
1
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2answers
42 views

$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
0
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1answer
13 views

For which value of “d” cantor set has lebesgue measure zero?

I tried to generalise the cantor set. I retain "$d<1$" size interval at both end and removed "1-2d" size interval, and process is same further. I have curious about what value of "d" this cantor ...
5
votes
1answer
47 views

What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
4
votes
1answer
31 views

Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$? [on hold]

Suppose $m$ is Lebesgue measure. Define $x + A = \{x + y : y \in A\}$ and $cA = \{cy : y \in A\}$ for $x \in \mathbb{R}$ and $c$ a real number. Let $A$ be a Lebesgue measurable set. I have two ...
3
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1answer
18 views

Example of Lebesgue measurable subsets satisfying conditions [on hold]

Let $m$ be Lebesgue measure. What is an example of Lebesgue measurable subsets $A_1, A_2, \ldots$ of $[0, 1]$ such that $m(A_n) > 0$ for each $n$, $m(A_n \Delta A_m) > 0$ if $n \neq m$, and $m(...
0
votes
1answer
37 views

What does a random variable 1 with subscript [0,1/2] mean?

I came across the following notation that I cannot follow: $1_{[0,1/2]}$ It is supposed to be some kind of random variable (or just an event? not sure) It is hard to google this, too. What does such ...
-2
votes
1answer
34 views

Meaning of measurable function [closed]

I don't understand the meaning of Measurable function , my lecture told us that $f(x)$ measurable on measurable set $E$ if $E(f>A)=\{x\in X:f(x)>A\}$ $1.\quad$Can you please give me examples ...
0
votes
1answer
54 views

Do we have $\|x\|_p \le \|x\|_q$, where $x$ is bounded and $p>q$? [closed]

Assume that $x\in L_p(0,\infty) \cap L_q(0,\infty)$, where $p>q$. Do we have $$\|x\|_p \le \|x\|_q?$$ where $x$ is bounded.
3
votes
2answers
56 views

What is the infimum/supremum of a set in the extended real line?

May be a simple question, but I still don't know what the definition of the supremum or infimum of a set in the extended real line is. For example when we define the Lebesgue measure, we define the ...
1
vote
1answer
20 views

Inferring absolute continuity of summands from absolute continuity of sum

Suppose we have i.i.d. random variables $X_1, X_2$. If $\text{Law}(X_1 + X_2)$ is absolutely continuous with respect to the Lebesgue measure $\lambda$, can we infer that each $X_i$ is absolutely ...
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votes
0answers
33 views

Hölder inequality application to show that f=1

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
4
votes
1answer
102 views

Conclusion about measurable functions from knowledge about continuous functions

Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb{R}$. We know that if $$\int f d\mu = \int f d\nu $$ for all continuous functions $f$ then $\mu=\nu$ and so the equation above holds for ...
2
votes
1answer
83 views

Find an appropriate function such that the composition is Lebesgue integrable on $[0,1]$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a measureable function. Show there exists an $\omega:\mathbb{R} \rightarrow\mathbb{R}$ such that $$\lim_{t \rightarrow \infty}\omega(t)=\infty$$ and $g(t)=\...
2
votes
2answers
37 views

Show the limit exists almost every where

Let $E\subseteq \mathbb{R}$ be a Lebesgue measurable set. Define $$f(x)=\operatorname{dist}(x,E)=\inf\{|x-e| \ : \ e\in E\}. $$ I want to prove that for almost every $x\in E$ that $$\lim_{r\...
0
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1answer
28 views

Lebesgue Measure, Continuous from above in $R^m$

This question is from a sample exam. How does one prove the measure of the set $A_r$ goes to $0$ as $r$ goes to $0$: Let $\mu$ be Lebesgue measure on $\mathbb{R}^m$, Let $A \subset \mathbb{R}^m$ ...
-1
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1answer
20 views

the relation between the sigma-algebras of two isomorphic spaces [closed]

It crosses my mind the following question : if X and Y are two isomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between $\...
6
votes
4answers
888 views

Is diameter of a set a measure?

Suppose the diameter of a nonempty set $A$ is defined as $$\sigma(A) := \sup_{x,y \in A} d(x,y)$$ where $d(x,y)$ is a metric. Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable ...
0
votes
0answers
5 views

On *triplets* of $\left[0,1\right]^{3}$, does injection keep measure?

When we have two sets $S_{1}$ and $S_{2}$ of real numbers, an injection $f$ from $S_{1}$ to $S_{2}$, and the Lebesgue measure $\mu$, we may easily have $\mu\left(S_{1}\right)>\mu\left(S_{2}\right)$....
1
vote
2answers
30 views

How do you rigorously explain the fact that $u \in L^p$ can be non defined over sets of measure 0?

In all the definitions of $L^p(\Omega)$ spaces I have been given these are defined to be the set of functions $f: \Omega \to \mathbb{R}$ whose norm $||\cdot||_{L^p}$ is finite. We define is as the ...
4
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1answer
50 views

Volume in higher dimensions

Let me first state the statement which I want to prove (encountered while studying "Geometry of Number"): Suppose $A$ is a convex, measurable, compact and centrally symmetric subset of $\...
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0answers
16 views

Lebesgue Theory in Higher dimensions

What are some things that we have to watch out for when going to higher dimensions (greater than or equal to 2) for Lebesgue Measure Theory? For instance, is there anything that is true in Lebesgue ...
1
vote
1answer
26 views

Approximation of Lebesgue Measurable Sets by compact and G-delta?

Just to check if this is true: Let $E$ be a Lebesgue measurable set on $\mathbb{R}^n$. Does there exists a sequence of open sets $G_m$, compact sets $K_m$, with $K_m\subset E\subset G_m$ and $\mu((\...
0
votes
1answer
13 views

Set used as a “mask” to “clip” that set

I am very interested to understand this phrase from wikipedia: "Intuitively, this condition means that the set E must not have some curious properties which causes a discrepancy in the measure of ...
0
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0answers
28 views

Inner lebesgue measure definitions

I´m currently studying lebesgue measure theory. I'm using Lebesgue integration on euclidean space by Frank Jones as one of my reference books. He defines the outer and inner lebesgue measures for an ...
6
votes
1answer
49 views

Example of set, finite outer measure, subsets, where outer measure does not converge

What is an example of a set $X$ and a finite outer measure $\mu^*$ on $X$, subsets $A_n \uparrow A$ of $X$, and subsets $B_n \downarrow B$ of $X$ such that $\mu^*(A_n)$ does not converge to $\mu^*(A)$ ...
5
votes
0answers
39 views

Does $\mu^*$ agree with $\mu$, measure space? [closed]

If $(X, \mathcal{A}, \mu)$ is a measure space, define$$\mu^*(A) = \inf\{\mu(B) : A \subset B,\,B \in \mathcal{A}\}$$for all subsets $A$ of $X$. I have a few questions? Is $\mu^*$ an outer measure? ...
10
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1answer
65 views

Non-invertible measure preserving transformations of $\mathbb{R}^n$

I am looking for particular examples of measure-preserving transformations of $\mathbb{R}^n$ (with Lebesgue measure) to get a better idea of how they behave. A large family of such transformations ...
2
votes
1answer
25 views

Continuity of a characteristic function of a translated set

Let $E \subseteq \mathbb{R}$ be a measurable set. Is it true that $\chi_{E+t}(x) \rightarrow \chi_{E}(x)$ as $t \rightarrow 0$, where $E+t = \{x+t \, | \, x \in E\}$ for each $t \in \mathbb{R}$, ...
4
votes
0answers
61 views

Show that $\int_{E}{F(x,t)}d\mu\otimes\lambda=\int_{X}{\int_{[\varphi_1(x),\varphi_2(x)]}{F(x,t)d\lambda(t)d\mu(x)}}$

Let $(X,\mathcal{F},\mu)$ be a $\sigma-$finite measure space. Let $\varphi_1,\varphi_2:X\to\mathbb{R}$ functions in $\mathcal{M}(X,\mathcal{F},\mathbb{R})$ such that $\varphi_1(x)\leq\varphi_2(x)$ for ...
0
votes
1answer
16 views

How should I refer to “volume-like” measures in a dimensional-free way?

When I try to model diffuse defined objects, I Frequently found myself in the need to refer to a measure in a way independent of the dimensional attributes of the space of definition of the objects. ...
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0answers
69 views

How to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$. First of all I need to check that both $f_n$ ...
3
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1answer
22 views

Exists $c \in \mathbb{R}$ where $A \cap (c + \mathbb{Q}) = \emptyset$, where $A$ has Lebesgue measure $0$

Suppose $A \subset \mathbb{R}$ has Lebesgue measure $0$. Does there exist $c \in \mathbb{R}$ such that $A \cap (c + \mathbb{Q}) = \emptyset$, where $c + \mathbb{Q} = \{c + x : x \in \mathbb{Q}\}$ and $...
3
votes
1answer
35 views

Does there exist $G$ open and $F$ closed such that $F \subset A \subset G$ and $m(G - F) < \epsilon$?

Let $m$ be Lebesgue measure and $A$ a Lebesgue measurable subset of $\mathbb{R}$ with $m(A) < \infty$. Let $\epsilon > 0$. Does there exist $G$ open and $F$ closed such that $F \subset A \subset ...
3
votes
1answer
22 views

Measure on Borel $\sigma$-algebra of $\mathbb{R}$ is Lebesgue-Stieltjes measure

Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0, x])$ if $x \ge 0$ and $\alpha(x) = -\mu((x, 0])$...
1
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1answer
73 views

Counterexamples in Analysis

I want to (dis)prove the following statement: A sequence of functions which converges almost uniformly implies uniform convergence for that sequence of functions. I'm sure I've read up on a ...
3
votes
2answers
52 views

Preimage of open set is Lebesgue measurable only if the function itself is measurable

It is a simple result in my book saying the proof is trivial, but I can not seem to show it. If someone can provide a hint just to help me begin my proof, it would be of assistance. Assume you know ...
2
votes
2answers
29 views

$\{A \subset X: \chi_A \in \mathcal{F}\}$ is a sigma algebra

Suppose $\mathcal{F}$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal{F}$ and $f + g$, $fg$, and $cf$ are in $\mathcal{F}$ whenever $f$, $g \in \...
0
votes
1answer
35 views

Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
0
votes
1answer
17 views

Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
1
vote
1answer
24 views

Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
2
votes
3answers
64 views

Is it correct to interpret the “dx” in the standard notation for integrals as the Lebesgue measure?

Ok so I am in my Calc I class for the summer and we are just beginning to talk about integrals. I know a little bit about measure theory and the Lebesgue integral and why is it more general than the ...
5
votes
1answer
123 views

Diffuse-like decomposition of the segment $[0,1]$ in accordance with Lebesgue measure

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any subsegment $[a,b]\...
1
vote
2answers
27 views

For any measurable set $A\subset\mathbb{R}$ and $r\in(0,\mu(A))$ we have $(\mu|_{2^A})^{-1}(r)\neq\emptyset$

Recently when I tried to prove a statement I needed to rely on the following fact that intuitively feels correct, but I wasn't able to prove it accurately. Here it is: Consider a set $A\subset\...
1
vote
1answer
31 views

Equivalent definition of Lebesgue measurability for sets?

When introducing measurability, we noted that we wanted the following property to hold for disjoint $A, B \in \mathcal{P}(\mathbb{R})$ $m(A \cup B) = m(A)+m(B)$ (additivity) We then defined a set A ...
2
votes
1answer
58 views

$m_*(E)=m^*(E)\iff E$ Lebesgue measurable

Let $E\subset [a,b]$. Show that $E$ is Lebesgue measurable if and only if the Lebesgue outer measure of $E$ is equal to the Lebesgue inner measure of $E$. I have seen the proof for this above ...