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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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Question about composition of measurable function f and continuous function g being measurable:

I have to show the following: Suppose $A$ is a measurable subset of $\mathbb{R}$ and $B$ an open subset of $\mathbb{R}$. If $f: A \to B$ is measurable, and $g:B \to \mathbb{R}$ continuous, then $g \...
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40 views

about unmeasurable set on R

Suppose $\{E_n\}$ is a sequence of sets in $\mathbb{R}$, and $E_n$ are pairwise disjoint sets satisfying $$ m^*(\cup_{n=1}^{\infty}E_n)<\sum_{n=1}^{\infty}m^*E_n. $$ I want to prove that there ...
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1answer
37 views

Showing Lebesgue Integral inequalities

Let $f,g: [0,1] \longrightarrow (0,\infty)$ be measurable and $\beta >0$. Assume that $$\int_{0}^{1}g(x)dx = 1.$$ Show that $$1\leq \Bigg(\int_{0}^{1}f(x)^{-\beta}g(x)dx\Bigg)\Bigg( \int_{0}^{1}f(x)...
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63 views

about Lebesgue measure on R

E is a set such that $m(E)>0$, $E \subset (0,1)$ and there exist $c>0$ such that for some moving interval $I$, $$\lim_{mI\rightarrow0}\frac{m(E\cap I)}{m(I)}=c$$ Proof:mE=1 My attempt ,I have ...
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0answers
31 views

Random variable defined on the Lebesgue probability space

There is a random variable X defined on the Lebesgue probability space whose cumulative distribution function is F. We can find X(w) knowing that: $X(ω)=\inf\{x∈R:F(x)>ω\}$. 1) how do we prove ...
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1answer
26 views

Find a function satisfies an integrals

Can we find an Lebesgue integrable function $f(x)$ satisfies the following integrals $\int_0^1 x^3f(x)= 2$ and $\int_0^1 (f(x))^4 dx = 125$ This is a qualifying question (measure theory) Or in ...
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1answer
51 views

For any set $E$ in $R$ and $k > 0 $, let $kE $= {$x : k^{-1}x ∈ E$}.How to show $m^*(kE) = km^*(E)$ and $E$ is measurable if and only if $kE$ is. [on hold]

I can intuitively feel by taking some examples that the equations in questions are true but i am not able to prove them.
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0answers
39 views

If f is differentiable in $[0,1]$, let $E_0=\{x\in [0,1]: f'(x)=0\}$, how to prove $m(f(E_0))=0$($m$ is the Lebesgue measure)?

If f is differentiable in $[0,1]$, let $E_0=\{x\in [0,1]: f'(x)=0\}$, how to prove $m(f(E_0))=0$($m$ is the Lebesgue measure)? I tried to use some little open intervals to cover every points in $E_0$,...
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Calculation of area of a disk directly from Lebesgue measure properties.

I'm currently stuck upon an apparently "easy" problem: calculate the area of a real disk using the theory of the Lebesgue measure in $\mathbb{R}^n$. I say "easy" because, using standard integration ...
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1answer
55 views

Let $m^\ast(E)$ < ∞.If for every interval (a, b) we have that $b-a$=$m^\ast((a,b)∩E) + m^\ast((a,b) ∩ E^c)$ then $E$ is lebesgue measurable

Since we are considering only few one type if sets (i.e open intervals) , i don't know how to prove $m^\ast(A)$=$m^\ast((A∩E) + m^\ast(A∩ E^c)$ using only information that $b-a$=$m^\ast((a,b)∩E) + m^\...
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1answer
46 views

Is there an invariant measure absolutely continuous wrt to the lebesgue measure for the map f

Let $f:[0,1]\rightarrow[0,1]$ where $f(x)=x/2$ $(1-x)$, and let $\lambda$ be the lebesgue measure on [0,1]. Is there a probability measure $\mu$ that is invariant and absolutely continuous wrt to the ...
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1answer
72 views

Examples of Lebesgue-integrable, but not Riemann-integrable functions

The standard example of this is the characteristic function of the rationals. However this is somewhat pathological as this function is zero almost everywhere. What are other examples that differ from ...
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1answer
18 views

Outer measure on $\mathbb R$ some sort of continuity about Measure

$m^*(E)=q>0$, for any $c\in (0,q)$, there exist $E_0\subset E$, such that $m^*(E_0)=c$ $$m^*(E)=\inf\{mG|E\subset G ,\text{G is open set}\}$$ I think if I can find a set $A\subset E$, and $m^*(E-...
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1answer
23 views

Lebesgue integration of a nowhere zero function. [closed]

Can the Lebesgue integration of nowhere zero (positive) function be zero over a set of non zero measure? Counter-example/Proof?
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1answer
12 views

Proof Check: Outer Regularity of Lebesgue Measure

I am trying to prove that for any bounded set $A$ in the borel $\sigma$ algebra, that for the Lebesgue measure $m$ $$ m(A)=\inf\{m(U)|U\text{ is open and }A\subseteq U\} $$ here is my attempt. Let $...
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0answers
9 views

How do you refer to $M$ points spanning up a less than $M-1$-dimensional Lebesgue measure?

Assume, for the sake of the argument, that I consider $M$ points in an $L>M$-dimensional space. These points can, at most, span up a $M-1$-dimensional object. However, it is also possible that ...
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1answer
31 views

Does $\lim_n n \int _c^{c+\frac {1 } {n }}f(x) \lambda(dx)=f(c)$ for bounded Lebesgue integrable $f $

As the title states I wonder if it is true, and if so how to show, that $\lim_n n \int _c^{c+\frac {1 } {n }}f(x) \lambda(dx)=f(c)$ for bounded Lebesgue integrable $f $. For a function continuous at $...
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1answer
38 views

Prove $\int_{|x| \ge \epsilon} \frac{dx}{|x|^{d+1}} = \frac{1}{\epsilon} \int_{|x| \ge 1} \frac{dx}{|x|^{d+1}}$

This is a textbook corollary in my Real Analysis text book. I'm slightly paraphrasing the first part that I fully understand: \begin{align*} f(x) &= \begin{cases} \frac{1}{|x|^{d+1}} &...
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1answer
26 views

Proof of measurability of set of all rationals [closed]

How we can prove that set of all rational numbers is measurable by showing that inner and outer measures are equal?
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19 views

Total variation and Dirac-delta function

I am trying to understand the following from Folland. In page 94 he states that the total variation of a complex measure $\mu$ is defined as the unique number $|\mu|$ such that $d|\mu| = |f| d v$ ...
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1answer
32 views

How to show convergence in $L^{p}$ spaces

Let $I =[1,\infty)$. Suppose $f: I \longrightarrow \mathbb{R}$ is measurable such that for some $\alpha >1$, $$|f(x)| \leq x^{-\alpha} \;\; \;\; (\forall x\in I).$$ For $n \in \mathbb{N}$, define $$...
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0answers
24 views

Is the Lebesgue measure of graph of every real function equal to zero?

The problem states: Find the measure of the set $\Gamma_f = \{(x, f(x)) \in \mathbb{R}^2 | x \in (0, 1]\}$, for $f : (0, 1] \rightarrow \mathbb{R}$, $f(x) = \frac1x$. There was a lemma which stated ...
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0answers
10 views

Can one construct a set $E$, $m^*(rE)\neq rm^*(E)$

Can one construct a set $E$, $m^*(rE)\neq rm^*(E)$. Well , we know that if $E$ is measurable then $$m^*(rE)= rm^*(E)$$,so what is the case of a none-measurable set ? Well,I think it holds for any ...
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1answer
22 views

If T preserves the measure of A and B then it preserves the Measure of $A \cap B$

Let $T : (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda^1) \rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda^1)$ be measurable. I claim that $T$ is measure preserving on $(\mathbb{R},\mathcal{B}(\...
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0answers
22 views

Interchanging limit and integral of difference quotient

I have shown that $$ T_h(x)= \begin{cases} x+h & x\in[0,1-h]\\ x+h-1& x\in(1-h,1] \end{cases} $$ for some fixed $h\in(0,1)$ is measurable and measure preserving on $([0,1],\mathcal{B}...
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1answer
36 views

Proving that a function is square integrable

How can one prove which of the following functions are $\in \mathcal{L}_2(-\infty,\infty)$? $f(x)=\frac{1}{1+x^2}$ $f(x)=\frac{1}{x^2}$ $f(x)=\frac{1}{1-x^2}$ I've run numerical simulations which ...
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1answer
33 views

$U \subset \mathbb{R}^d$ open and $D \subset U$ open and dense $\implies \lambda(D) = \lambda(U)$

This is not a homework exercise! Let $U \subset \mathbb{R}^d$ be and open subset and $D \subset U$ open and dense in $U$. Can we conclude that $\lambda(D) = \lambda(U)$? Here, $\lambda$ denotes ...
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3answers
37 views

if $f\phi \in L^p(\mathbb{R})$ for every $f \in L^p(\mathbb{R})$, then $\phi \in L^{\infty}(\mathbb{R})$.

Let $\phi: \mathbb{R} \to \mathbb{C}$ be a measurable function and fix $1 \leq p \leq \infty $. Show if $f\phi \in L^p(\mathbb{R})$ for every $f \in L^p(\mathbb{R})$, then $\phi \in L^{\infty}(\mathbb{...
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Intuition behind the definition of Lebesgue measurable function

In Real Analysis written by Royden, the definition of measurable function is as follows. An extended real-valued function f defined on E is said to be Lebesgue measurable, or simply measurable, ...
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1answer
18 views

Compact subset and almost everywhere limit

Let $A$ be a compact subset of $\mathbb{R}$ of positive Lebesgue measure. Assume that almost everywhere the limit $f(x)=\lim_{\delta\to 0}\frac{\lambda(A\cap (x-\delta,x+\delta))}{2\delta}$ exists, ...
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1answer
16 views

Supremum of product $L^{\infty}$ functions [closed]

Let $ f $ and $g $ be $ L^{\infty}(R^n) $ functions. It's true that $$ \sup\left( f\cdot g\right) = \sup f \cdot \sup g \ ? $$ Can anyone could me explain why?
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1answer
31 views

Cantor-like Set not of measure zero

I have a short question, but that keeps me stuck for a couple of days. Let's start saying that Cantor Set is defined as: $$C:=\left\{x\in\mathbb{R}|x=\sum_{n\in\mathbb{N}}\frac{\alpha_n}{3^n}, \...
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2answers
32 views

$d(x,E^c) > \frac{1}{j}$ is compact if $E$ is open.

It is claimed in an analysis text that Let $E \subseteq \Bbb R^n$ be an open set. Then $$K_j := \{ x \, :\, d(x, E^c) \ge 1/j \}$$ is a compact set. How does one see this? I guess it is ...
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31 views

Transformation between two measures

If $\mu$ and $\nu$ are two measures, both absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^d$ with smooth densities $p_\mu(\mathbb{x})$ and $p_\nu(\mathbf{x})$, does it always ...
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0answers
20 views

Do injective outer measure-preserving functions preserve measurable sets?

Suppose $f:\mathbb{R}^n\to\mathbb{R}^n$ is an injection, such that for any $E\subseteq\mathbb{R}^n$ we have $m^*(E)=m^*(f(E))$. Is it true that for any $E\subseteq\mathbb{R}^n$ measurable, $f(E)$ is ...
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0answers
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Question about Lemma 3.15 in Real Analysis; Modern Techniques and Their Applications by Folland, G. B.,

Lemma. Let $C$ be a collection of open balls in $\mathbb{R}^n$, and let $U =\cup_{B\in C} B$. If $c<m(U)$, there exist disjoint $B_1,...,B_k\in C$ such that $\sum_{k=1}^{k}m(B_j)> 3^{-n}c$. ...
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31 views

Cardinality of the set of bijective Lebesgue-measurable functions on the unit intervall

Let $I= [0,1]$ be the unit intervall. It is well known that the cardinality of $$B = \{ f: I \to I \mid f \text{ bijective} \}$$ and $$L = \{ f: I \to I \mid f \text{ Lebesgue-measurable} \}$$ is $2^{|...
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Lebesgue Measure Question - Let $T$ be Lebesgue measurable set:

Suppose $A \subset T$ and $B \cap T = \emptyset$. Show that $m^*(A \cup B) = m*(A)+m*(B)$. My thoughts: I know that $m^*(A \cup B) \leq m^*(A)+m^*(B)$ by having already proven this, so it will be ...
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2answers
30 views

Tips to show $\lim_{n \to \infty} \int_{X}|f|^{\frac{1}{n}}d\mu=\mu(\{f\neq0\})$

Let $(X,\mathcal{A}, \mu)$ be of finite measure and $f$ integrable on the measure space. Show that: $\lim_{n \to \infty} \int_{X}|f|^{\frac{1}{n}}d\mu=\mu(\{f\neq0\})$ My ideas: I initially ...
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1answer
26 views

Let $g: \mathbb{R} \to \overline{\mathbb{R}}$ be integrable. If $\int_K g \, d m = 0$ for every compact $K \subset \mathbb{R}$, then $g = 0$ a.e.

I need to prove the statement: Let $g: \mathbb{R} \to \overline{\mathbb{R}}$ be Lebesgue integrable. If $\int_K g \, d m = 0$ for every compact $K \subset \mathbb{R}$, then $g = 0$ a.e. Note that $...
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2answers
64 views

Prove that there exist a compact set such that $m(E\cap K^c)<\epsilon$

Let $E\subset\mathbb R$ Lebesgue measure with $m(E)<\infty$. Prove that for each $\epsilon>0$, there exist a compact $K\subset E$ such that $m(E\cap K^c)<\epsilon.$ Attempt By contradiction,...
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1answer
48 views

Does there exist a compactly supported integrable function with infinite Coulomb energy?

The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that $$ E[f] = \iint\limits_{\Omega\...
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1answer
30 views

Approximate the integral of an unsigned measurable function

I've been struggling with this problem. Let $f\colon X \to [0,+\infty]$ be an unsigned measurable function. Suppose $\int f < \infty$. Prove that for every $\epsilon > 0$ there exists a set $E ...
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1answer
47 views

Hahn Decomposition Theorem In Folland

I was reading the proof of Hahn Decomposition theorem from the textbook of Folland: precisely I was looking at the following text I have the following question: As Highlighted in the text above, why ...
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1answer
52 views

$ f\in C_{0}^{\infty} \Rightarrow f\in L^m $?

Let $ C_{0}^{\infty} $ be the subspace of $ C^{\infty} $ functions with compact support in $ R^n $. It's true that if $ f \in C_{0}^{\infty}, $ then $ f \in L^{m} \cap H^{s} $ where $ 1\leq m \leq 2$ ...
4
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1answer
115 views

Existence of a particular inverse transformation

Let $h : \mathbb{R}^D \rightarrow \mathbb{R}^d$, where $d < D$, be a differentiable function. I would like to find minimal conditions under which there exists a differentiable function $g : \mathbb{...
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0answers
8 views

Measure convex hull point and ball

The Lebesgue's measure (in $R^{n}$) of the convex hull of $B(0,a)$ and a point $p$ is $C_{n} a^{n-1}|p|$ with $C_{n}$ constant depending only on the dimension $n$. I don't know how to prove it, I ...
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0answers
25 views

Proof of translation and linear transformation invariance of R_k

Let A be a Lebesgue measurable set. Suppose that $f$ and $g_r$ are respectively the translation and linear tansformation with ratio $r$ on $\mathbb{R}_k$. Prove that $f(A)$ and $g_r(A)$ are Lebesgue ...
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1answer
21 views

A question about measure of symmetric difference of sets

Suppose that $\mu$ is a measure and sigma-algebra $\Sigma$ over $X$. Suppose that $0<\mu(A)<\infty$ and for every $E\in \Sigma$ and $0<\mu(E)$ then $\mu(A\Delta E)=0$. If we consider $L^p$ ...
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1answer
46 views

Define function and image of fat Cantor set

This construction I found it paper published in $1965$ I think. Here is the way that defined. Let $I=[0,1]$ and define a Cantor set as follows. $C_1$ obtained from $I$ by taking the open interval ...