Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

learn more… | top users | synonyms (1)

4
votes
1answer
135 views

Does there exist a subset with given inner- and outer Lebesgue measures?

Is it possible for every $0\leq a< b \leq \infty$ to find a set $A\subset R^n$ such that the inner Lebesgue measure of $A$ is equal $a$ and the outer Lebesgue measure is equal $b$. It is true ...
1
vote
1answer
171 views

set of values of finite measure, exhaustion method

Let $\cal{S}$ be a $\sigma$-field of subsets of a set $Z$, and $\mu$ be a positive finite measure on $\cal{S}$ which does not contains atoms. (An atom in $(X, \cal{S}, \mu)$ is a measurable set $E$ ...
1
vote
1answer
213 views

Measurablity of inner product

If $f : X \to \mathbb{R}^n$ is measurable, then $\langle f,f\rangle = ||f||_2^2: X \to \mathbb{R}$ is measurable (if $\langle f,f\rangle < c$ for $c > 0$ then $f$ should lie in an open ball with ...
6
votes
2answers
266 views

Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?

I want to show that if $f$ is non-increasing and $f\in L_{1}([a,\infty),m)$ where $m$ is Lebesgue measure then $\lim_{t\to\infty} t f(t)=0$. So far I've been able to show that $f\geq 0$ and that ...
4
votes
1answer
192 views

Problem about absolute continuity of a function

$f:\mathbf{R} \to \mathbf{R}$ is an increasing function with $\lim_{x\to -\infty}f=0$ ,$\lim_{x\to \infty}f=1$, and $\int_{R}f'=1$. Prove that $f$ is absolutely continuous on every interval ...
2
votes
2answers
267 views

Detail in Conditional expectation on more than one random variable

I have $E(X|Y,Z)=0$, $X$ independent of $Y$ and of $Z$ and I want to conclude that $E(X)=0$ ($X,Y,Z$ are real-valued random variables). Okay it seems quite obvious, but if I try to make a strict ...
1
vote
1answer
155 views

Uniformly continuous $f$ in $L^p([0,\infty))$

Assume that $1\leq p < \infty $, $f \in L^p([0,\infty))$, and $f$ is uniformly continuous. Prove that $\lim_{x \to \infty} f(x) = 0$ .
55
votes
4answers
3k views

Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
1
vote
1answer
215 views

Finiteness of conditional expectation if expectation is finite

I have $E(X) < \infty$. Under which conditions follows that $E(X|A)<\infty$ ? (A is an event of the form {$Y=y$} if it should matter) If I can use the formula $E(X|A)=\frac{E(X 1_A)}{P(A)}$ ...
4
votes
0answers
90 views

A detail about MCT application

I have a indirect question about Monotone Class Theorem (MCT), in its functional form. Here is a version which should be sufficiently general for my purpose. Functional Monotone Class Theorem : ...
16
votes
1answer
757 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
3
votes
1answer
234 views

Decomposing a Bounded Linear Functional on Lp as a difference of Positive Bounded Linear functionals

I am learning Measure theory via self study of Bartle "The elements of Integration and Lebesgue Measure". I was stumped by the reasoning in one of the decomposition proofs. The point is to show that a ...
9
votes
0answers
285 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
2
votes
3answers
339 views

How can a density be larger than $1$?

From Frank Morgan: Geometric Measure Theory, Fourth Edition: A Beginner's Guide, page 13,the $2$-dimensional density of the cone $x^2+y^2=z^2$ at $0$ is $\sqrt{2}$. I feel strange of that,roughly ...
17
votes
3answers
3k views

The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$

I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. ...
6
votes
2answers
653 views

Lebesgue outer measure of $[0,1]\cap\mathbb{Q}$

Consider the Lebesgue outer measure $$ \bar{m}(X) = \inf_{A \supset X}\bigg\{\sup_{P\subset A}\quad m(P)\bigg\} $$ where $X = [0,1]\cap \mathbb{Q}$ and $P = \bigcup [a_i,b_i]$ is a suitable union ...
8
votes
2answers
483 views

Haar's base for $L^2[0,1]$

$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ ...
7
votes
1answer
200 views

Fixed point: sets and measures

Let $X$ be a Borel space with a Borel measure $\mu$. Suppose $\xi: X\times X\to\mathbb R_{\geq 0}$ is a continuous function and put $s(x) = \{y\in X:\xi(x,y) = 0\}$. For any set $b\in\mathcal B(X)$ we ...
4
votes
1answer
177 views

Approximating convex sets with disjoint rectangles in an optimal way

Let $O \subset \mathbb{R}^2$ be a convex open set of finite Lebesgue measure $1=m(O)$. Let's call a collection $P$ of $n$ disjoint open rectangles contained in $O$ a "partial cover of $n$ pieces". ...
1
vote
1answer
158 views

Restriction of measure to rationals

Let $X = [0,1]$ and $\mathbb Q$ - the set of rational numbers. We take $X' = X\cap \mathbb Q$ and define a measure on it such that $\lambda(X'\cap (a,b)) = b-a$ for any $a,b\in X$. This ...
6
votes
1answer
306 views

Hahn-Banach to extend to the Lebesgue Measure

I remember reading an example in a textbook that went something like this: if we take a function $\ell(f) = \int_{0}^{1}f(t)\, dt$, (with this being the Riemann integral) defined only on the set of ...
32
votes
3answers
3k views

How do people apply the Lebesgue integration theory?

This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas. In a lot of calculus textbooks, there is ...
7
votes
2answers
356 views

Carathéodory's method gives a complete measure

I would really like to show that the following is true. "Suppose that $X$ is a set and $\theta$ is an outer measure on $X$, and let $\mu$ be the measure on $X$ defined by Carathéodory's method. Then ...
3
votes
2answers
317 views

Measure of Image of Linear Map

I am trying to work my way through the proof of the change of variables theorem for Lebesgue integrals. A key lemma in this context is as follows: If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a ...
8
votes
2answers
3k views

Is there an example of a sigma algebra that is not a topology?

Is there an example of a sigma algebra that is not a topology? If this is not the case, is it possible to prove that all sigma algebras are topologies?
4
votes
1answer
303 views

Something connected with Ulam's tightness theorem

Well known theorem of Ulam says, that each probability measure $\mu$ defined on Borel subsets of polish space $X$ satisfies the following condition: for each $\epsilon>0$ there is a compact subset ...
4
votes
2answers
175 views

Coincidence criterion (measure theory)

The following theorem has been mentioned (and partially proved) in the book Functions of Bounded Variation and Free Discontinuity Problems by Luigi Ambrosio et. al. Let $\mu,\nu$ be positive measures ...
2
votes
0answers
235 views

Is the product of Borel spaces a Borel space?

Let $(S_i,\mathbf{S}_i)$ be a sequence of Borel spaces, i.e. such that for all $i$ there is a 1-1 bimeasurable map $\varphi_i:S_i\to T_i$, where $T_i$ is a Borel subset of [0,1]. Is $\prod_{i=1}^n ...
2
votes
1answer
254 views

An exercise for weak convergence

Recently, I found an exercise in Hunter's Applied Analysis(last page in the link), which may be closely related to the question I raised two months ago. Consider heat flow in a rod with rapidly ...
6
votes
4answers
264 views

Convergence of a kind of difference quotient

Let $f:\mathbb{R}\to \mathbb{R}$ be a measurable function and $(h_j)_j$ be a sequence of nonzero real numbers converging to zero as $j\to \infty$. Is it true that for almost every $x\in \mathbb{R}$: ...
3
votes
1answer
93 views

Random variable problem

Define a discrete random variable. Let $(Ω, A, P )$ be a probability space with $Ω = \{1,2,3,4,5,6\}$ and $F = \{Φ, \{1,3,5\}, \{2,4,6\}, Ω\}$. Define functions $X$, $Y$, $Z$ on $Ω$ as $X(k)= k$, ...
33
votes
2answers
2k views

Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject(single-variable calculus): the ...
10
votes
3answers
664 views

Ash's construction of the Lebesgue-Stieltjes Measure from a distribution function

I'm reading this book Probability & Measure Theory by Ash. I think I've come across a part that is a little hand-wavy. We are trying to build a Lebesgue-Stieltjes measure from a distribution ...
1
vote
1answer
99 views

Absolute continuity of the kernel

Let $(X,d)$ be a metric space and $K:X\times \mathcal B(X)\to[0,1]$ is a stochastic kernel on $X$. We call this kernel absolute continuous with respect to a measure $\mu:\mathcal B(X)\to\mathbb ...
2
votes
2answers
199 views

Set of measure zero

Let $(X,\mathcal F,\mu)$ be a measure space. For a measurable function $f:X\to\mathbb R$ define $S = \{x:f(x)>0\}$. Suppose for some set $B\in\mathcal F$ holds $$ \int\limits_{S\cap B}f(x)\mu(dx) ...
3
votes
1answer
787 views

Approximation by a $G_{\delta}$ set in Outer Measure implies Measurability

Question: If $E\subseteq {\mathbb R}$ and if there is some $G_{\delta}$ set $G$ such that $E\subseteq G$ and $m^{\ast}(G - E) = 0$ (where this is the outer measure that is used to define the Lebesgue ...
3
votes
0answers
86 views

Notation for a certain kind of discrete measure

Suppose $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ is smooth, $Z=\{x: \phi(x)=0\}$ and $D\phi\neq0$ on $Z$. Is anyone familiar with use of the notation $dZ$ for the measure $$\sum_{x \in Z} ...
5
votes
1answer
362 views

Nested sets convergence

Define $\xi\in C^1([-1,1]\times[-1,1])$ such that $$ \int\limits_{-1}^1 \xi(x,y)\,dy = 1 $$ for all $x\in[-1,1]$ and $\xi\geq 0$. Put $A_0 = [0,1]$ and $$A_{n+1} = \left\{x\in ...
2
votes
1answer
212 views

A calculation involving the normalized area measure

I am reading about the Dirichlet Space right now. The definition of a Dirichlet space is the set of all holomorphic functions in the unit disc that are finite with respect to the semi-norm: $\mid \mid ...
14
votes
3answers
480 views

Weird subfields of $\Bbb{R}$

I found this problem, and I can't get an answer to it: Prove that there are subfields of $\Bbb{R}$ that are a) non-measurable. b) of measure zero and continuum cardinality. I can't ...
2
votes
0answers
56 views

How can I prove that functions in a certain family have unique maxima?

I'm working on a problem in game theory, and I've run into a proposition that I don't know how to prove. Leaving out the game theory bit, I have a family of functions defined over a k-dimensional ...
0
votes
2answers
107 views

finite measures

$F$ is a finite measure on $(X,A)$ $a$ and $b$ belong to $A$ show that $F(a \cup b)=F(a)+F(b)-F(a \cap b)$ I have no idea how to approach this question. Any assistance would be appreciated.
23
votes
9answers
5k views

Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the ...
4
votes
5answers
459 views

Non-measurable Set

Suppose we define an equivalence relation on $[0,1)$ by saying that $x \sim y$ iff $x-y$ is rational. Let $N \subset [0,1)$ which contains exactly one member from each equivalence class. Also let $R = ...
4
votes
1answer
119 views

Closure of an invariant set

Consider a complete metric compact space $X$. For each $x\in X$ we define a probability measure $T(\cdot|x)$ over a Borel sigma-algebra $\mathcal{B}(X)$. We call a set $A\subset X$ invariant if ...
1
vote
1answer
440 views

Density of measure

Let $dx$ be the Lebesgue measure on $\mathbb R^d$. Let $u:\mathbb R^d\to{\mathbb R}\cup\{\infty\}$ is a non-negative and measurable function. The question is that, what are the conditions on $u$ so ...
3
votes
3answers
734 views

the pseudometric induced by a measure

Let $(X, \Sigma, \mu)$ be a measure space. We can define a pseudometric $d$ on $\Sigma$ in the following way: $$d(A, B) = \mu(A\bigtriangleup{}B)$$ where $A\bigtriangleup{}B = ...
3
votes
1answer
191 views

Boundary of a Borel set

Let $X$ be a topological space and $B$ is a Borel set of this space, i.e. $B\in\mathcal{B}(X)$ where $\mathcal{B}(X)$ is the smallest $\sigma$-algebra which contains all open subsets of $X$. Let ...
0
votes
1answer
135 views

sequence of step maps which converge almost everywhere

I am trying to learn some measure theory from Lang's Real and Functional Analysis and am having difficulty understanding a claim he makes without proof. Let $(X, \scr{M}, \mu)$ be a measured space, ...
5
votes
1answer
301 views

Measure of the boundary

Let us been given a bounded domain $A\subset \mathbb{R}^n$. There is a function $u:A\to[0,1]$ such that $$ A' = \{x\in A:u(x) = 1\} $$ is not empty and $u\in Lip(A)$ with rate $\alpha$. Is it ...