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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2
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1answer
27 views

Please check whether the proof is correct or not.

Please check my solving. I want to know where to be wrong or illogical, or where logical jumps are. Problem Let $y=Tx$ be a nonsingular linear transformation of $\mathbb{R}^n$. If ...
0
votes
0answers
28 views

$f'$ Lebesgue-integrable

Let $f:[a,b]\to\mathbb R$ be differentiable and the derivative $f'$ bounded. How to show that $f'$ is Lebesgue-integrable on $[a,b]$ and $$\int_{[a,b]}f'd\mu=f(b)-f(a)$$ where $\mu$ denotes the ...
0
votes
1answer
17 views

When summation of two sequences is finite, is one finite?

$|\cdot|$ is Lebesgue measure. Let $w(\alpha) := |\{x:f(x)>\alpha\}|$ Let $f$ be a nonnegative function. Then, the proof uses that $\displaystyle\sum_{k=-\infty}^{\infty} 2^{kp}w(2^k)\lt\infty ...
1
vote
1answer
18 views

For $E \subset \mathbb{R^n}$, is $f^p$ finite almost everywhere in $E$ if $f \in L^p(E)$?

Q1) I didn't learn $L^p(E)$ yet, only learned $L(E)$. In order to solve problems, however, I need to know that the following theorem is correct or not. Let $E \subset \mathbb{R^n}$. Then, if $f \in ...
0
votes
0answers
13 views

Lebesgue-$\sigma$-algebras $\mathfrak L^{p+q}\neq\mathfrak L^p \otimes\mathfrak L^q$

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...
1
vote
1answer
48 views

Dimensions of a sphere and a ball

The volume of the unit ball in $\mathbb{R}^n$ is denoted by $v_{n}$ and the surface area of the unit sphere $S^{n-1}$ is denoted by $\omega_{n-1}$. What is the importance of writing $n-1$ and $n$?
1
vote
1answer
25 views

Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$.

Problem: Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$. Attempt: Well, if $f \equiv 0$ we get 1. Provided some sort of goodness like $f \in C^1$ ...
0
votes
2answers
36 views

Sequence of Radon Measures $\mu_n$ on $\mathbb{R}$

Problem: Find a sequence of signed Radon Measures $\mu_n$ on $\mathbb R$ such that $\langle \mu_n, \phi \rangle \to 0$ for every $\phi \in C^1_c(\mathbb R)$, and $|\mu_n|([0,1]) \to +\infty$. ...
3
votes
0answers
30 views

minimize the area of convex hull of sum of 3 balls

How should we place 3 balls $B_1,B_2,B_3$ on the plane, if we want to minimize the area of convex hull of $B_1\cup B_2\cup B_3$ ? Balls can have boundary common points only -- the intersection of any ...
0
votes
1answer
25 views

$\mathcal{L}^N(B_r(x)\cap E)> 0 \hspace{0.6cm} \forall r>0$ if every point is a Lebesgue Point

Exercise: Let $E$ be a Borel set such that every point is a Lebesgue Point for $\chi_E$ , and let $x \in \partial E$ (the topological boundary). Show that $\mathcal{L}^N(B_r(x)\cap E)> 0$, and ...
0
votes
1answer
40 views

Can someone solve my non-understandable process in proving a theorem?

Theorem. Let $E$ be a subset of $\mathbb{R}^n$. Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$. For your ...
0
votes
0answers
16 views

Interesting measure theory property in L^p [duplicate]

Let $f, f_n \in L^p (X)$, so that there is a function $g\in L^p (X)$ with $|f_n|\leq g,\ \forall n$ and $\forall \epsilon>0, \lim_{n\to\infty} \mu (\{x\in X\big | |f_n (x)-f(x)|\geq \epsilon\})=0$. ...
0
votes
1answer
26 views

Product of Lebesgue-null-set and arbitrary Lesbesgue-set is a Lebesgue-null-set again

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...
0
votes
1answer
30 views

Lebesgue integral, path connected and compact function

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
1
vote
1answer
17 views

$A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$

Let A be a real set then is it true that $A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$.
-1
votes
0answers
34 views

Convergence of $\chi_{A_n}$ to $\chi_A$, where $A$ is the union of the sets $A_n$ [on hold]

Suppose $(X,\mathcal{M},\mu)$ is a measure space with $\mu$ is a complete measure. Let $f$ be a measurable function on measurable subset $A$ of $X$. Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of ...
1
vote
0answers
22 views

Exercise 8.O in Bartle's The Elements of Integration

I have a doubt about this exercise (8.O) in Bartle's book. Exercise 8.O I already answered the Exercise 8.N so I'm able to apply it, but, I just have no idea about how to do this. I'm working on ...
2
votes
1answer
27 views

Prove that summation of nonnegative and measurable $f_k$'s is summation of each integral of $f_k$.

Prove. If $f_k$, $k=1, 2, \cdots,$ are nonnegative, measurable, and defined on $E\subset\mathbb{R}^n$, then $$\int_E{\left(\sum_{k=1}^{\infty}f_k\right)}=\sum_{k=1}^{\infty}\int_E{f_k}$$ Proof. ...
0
votes
1answer
18 views

Continuity of Integration (Lebesgue)

On the theorem regarding continuity of integration: Let $f$ be integrable over $E$. If $\{E_{n}\}^{\infty}_{n=1}$ is an ascending countable collection of measurable subsets on $E$, then ...
0
votes
1answer
19 views

Let $A,B \subset \mathbb R$ with $m^*(A)=0$. Prove $m^*(A \cup B)=m^*(B).$

Let $A,B \subset \mathbb R$ with $m^*(A)=0$. Prove $m^*(A \cup B)=m^*(B).$ $m^*(A \cup B)=m^*(A) + m^*(B)$ if A and B are disjoint. In general, $$m^*(A \cup B) \leq m^*(A) + ...
1
vote
0answers
24 views

finding the Lebesgue measure of the set $A = \{ (x,y,z) \in R^3 | y^2 + z^2 \le 4, 0 \le x \le 4, x \le 6-y-z \}$.

I want to find the Lebesgue measure, denoted $u$, of this (closed and thus measurable) set. $$A = \{ (x,y,z) \in R^3 | y^2 + z^2 \le 4, 0 \le x \le 4, x \le 6-y-z \}$$ This is a cylinder cut by a ...
1
vote
0answers
16 views

algebraic sum of a graph of continuous function and itself - measure > 0 imply nonempty interior?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Then $G+G$ is compact: algebraic sum of a graph of continuous function and itself Borel or ...
2
votes
1answer
30 views

Why the probability distribution of a uniform random variable is the Lebesgue measure?

Consider the random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ distributed as a uniform on $[0,1]$. The probability distribution function of $X$ is defined as a map $$ ...
1
vote
0answers
35 views

Does $m^*(E)=m(E\cap A)+m^*(E\cap A^c)$ always true?

Caratheodory says that $E\subset \mathbb R^d$ is measurable if $$m^*(A)=m^*(E\cap A)+m^*(E^c\cap A)$$ for all $A\subset \mathbb R^d$. To me a more natural way would have been to define it as : $E$ is ...
1
vote
0answers
54 views

Comparison test and DCT

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
0
votes
1answer
58 views

$f$ integrable iff $\sum_{n=1}^{\infty} f(n)$ converges absolutely

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
5
votes
1answer
61 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
2
votes
1answer
57 views

Sum of measurable functions is measurable: countable choice required?

The standard proof that the sum of measurable functions is measurable uses countable choice, via the countable subadditivity of outer measure ($\implies$ measurable sets are closed under countable ...
1
vote
1answer
57 views

$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$

How can I prove $$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$$ given a measure space $(\Omega,\mathfrak A, \mu)$, a non-decreasing sequence $(f_n)$ of measurable functions on ...
0
votes
0answers
32 views

Is the composition of uniformly distributed functions uniformly distributed?

Let $\mathcal{I}:=[0,1]$. Def: A measurable function $\varphi:\mathcal{I}\rightarrow \mathcal{I}$ is said to be uniformly distributed with respect to the Lebesgue measure $\Lambda$ if, for any ...
-3
votes
0answers
38 views

calculation of Lebesgue measure [closed]

Which of the following subsets of $\mathbb{R}^2$ have positive Lebesgue measure? unit circle $U=\{(x,y):x=y\}+\{(x,y):x=-y\}$ $S=\{(x,y):x^2+y^2<1\}$ $T=\{(x,y):x=y\}+\{(x,y):x=y\}$ May be ...
2
votes
1answer
27 views

The composition of measurable function is not measurable: only for Lebesgue-measurability?

Let $\mathcal{I}:=[0,1]$. Let $\mathcal{R}(f)$ denote the range of a function $f$. Let $\Sigma$ be the $\sigma$-algebra of $\mathcal{I}$. Consider the measurable and continuous functions ...
0
votes
1answer
34 views

Using the monotone convergence theorem to show a function is integrable

Apply the monotone convergence theorem and the fundamental theorem of calculus to show that $f(x) = \left\{ \begin{array}{ll} x^{-a} & \mbox{if } 0 < x \leq 1 \\ \infty & \mbox{if } ...
0
votes
1answer
21 views

algebraic sum of a graph of continuous function and itself Borel or measurable?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Does $G+G$ have to be: a Borel set? Lebesgue measurable?
1
vote
1answer
23 views

regularity of a measure

Let $\mathcal{A}$ be a $\sigma$-algebra containing the Borel algebra (everything is in a topological space). Let $m\colon\mathcal{A}\to[0,\infty]$ be a measure. The standard definition of regularity ...
1
vote
0answers
22 views

In which cases the outer Lebesgue measure is additive?

In which cases the outer Lebesgue measure is additive? it is known that $m^*(A\cup B)=m^*(A)+m^*(B)$ holds for disjoint bounded closed sets. 1) is it true for any disjoint closed sets? 2) is it ...
1
vote
1answer
42 views

Integral of $f'$ where $f$ is continuous on $[a,b]$ and differentiable over $(a,b)$.

There is a problem which states that if $f$ is a function continuous on $[a,b]$ and differentiable almost everywhere on $(a,b)$ whose $|\text{Diff}_\frac{1}{n} f| \leq g$ almost everywhere on $[a,b]$ ...
0
votes
2answers
62 views
+50

intersection of boundaries have Lebesgue measure 0

Suppose $A$ and $B$ are two sets in $R^n$ such that $\overline{A}\cap B \cup \overline{B}\cap A$ is empty then $\partial A \cap \partial B$ has $n$-dimensional Lebesgue measure $0$ (where $\partial ...
1
vote
0answers
8 views

Jordan measure: half open intervals versus closed intervals.

I'm reading the definition of Jordan measure here and I have problems in understanding why the main definition should be given considering $n$-dimensional rectangles given by half-open intervals, ...
0
votes
1answer
25 views

Prove that $f^*(x)\geq \frac{c}{\|x\|^n}$ if $f$ integrable.

Let $f:\mathbb R^n\longrightarrow \mathbb R$ a non-zera integrable function. Set $$f^*(x)\geq \sup_{B\ni x}\frac{1}{|B|}\int_B f(y)\mathrm d y,$$ where the supremum is taken over all the ball that ...
0
votes
0answers
19 views

Measurability and integrability of set and function

My textbook said: Let $E\subset\mathbb{R}^n$, let $G$ be an open set, and let $|\cdot|_e$ denote outer measure. if $\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given ...
1
vote
3answers
44 views

$f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$

If $f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$. I know that if $E$ is of finite measure, then we can infer from $f(x)\in L^2(E)$ to get $f(x)\in L^1(E)$. However, now ...
0
votes
0answers
39 views

Find the Lebesgue measure of the following sets.

Find the Lebesgue measure of the following sets: i) A=$(\cup_{n=1}^\infty [2^n, 2^n + \frac{1}{2^n}))$ \ $\mathbb{Z}$ ii) B=$(\cup_{n=1}^\infty (n^n, n^n + \frac{1}{2^n}))$ $\cap$ $\mathbb{Q}$. For ...
-1
votes
0answers
34 views

Prove that h(x) is measurable and calculate the integral of h [closed]

Let $A,B\subset \mathbb {R}$ Lebesgue-measurable sets. I want to prove that the function $h(x)=|(A-x)\cap B)|$ is measurable and that $\int_{\mathbb {R}}h(x)dx=|A|.|B|$, where $|•|$ is the ...
2
votes
1answer
37 views

Let $f$ be positive and Lebesgue measurable on $[0,1]$. Show that $\inf_{\lambda(E)\geq \epsilon} \int_E fd\lambda >0$ for any $\epsilon\in(0,1]$.

The title says it all. I've already shown, for an earlier part of this problem, that for any $E$ with $\lambda(E)>0$, we have $\int_E fd\lambda >0$. I did that by reductio, showing that ...
2
votes
1answer
38 views

Can the unit interval be the disjoint union of countably many “super-dense” parts?

I'm curious about this question in the case where $f$ is not necessarily measurable. I think what it comes down to is this: Is there an $\varepsilon<1$ and a partition of $[0,1]$ in countably ...
-4
votes
0answers
17 views

Property about Lebesgue Integral for Bounded measurable function [closed]

[![property][1]][1] f and g bounded measurable functions defined on a set E of finite measure. How can I prove that property? Characteristic function of E1 union E2 is the sum of characteristic ...
1
vote
1answer
48 views

Is every set of measure zero countable?

I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable?
0
votes
0answers
15 views

Prove that $\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y))$ is measurable.

If I have two measurable functions $f:\mathbb R^p\to\mathbb R^m$ and $f:\mathbb R^q\to\mathbb R^n$, how can I prove that $$\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y))$$ is ...
2
votes
1answer
36 views

If $f$ is finite almost everywhere and $E$ is of finite Lebesgue measure, then f is “almost” bounded

If a real measurable function $f : E \rightarrow \mathbb{R}\ \cup \{\infty\}$ is finite almost everywhere (i.e. $f^{-1}(\{\infty\})$ has measure zero) and $E \subset \mathbb{R}$ is of finite Lebesgue ...