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

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6
votes
2answers
956 views

Lebesgue's Density Theorem - intuition and weaker forms

Lebesgue's Density Theorem states that given a measurable set $E$ on the real line, then the set of points $E'$ for which $\lim_{h \to 0} \frac{m(E \cap (x-h,x+h) )}{2h} = 1$ is $E$ up to a ...
3
votes
1answer
77 views

Is it possible to restrict an “almost” bijective measurable function on $(0,1)$ to a bijection?

Let $f:(0,1)\rightarrow (0,1)$ be a borel measurable function such that for every $y$ in $(0,1)$ , $f^{-1}(y)$ is a borel set and $\mu(f^{-1}(y))=0$ and also $\mu (f((0,1)))=1$ where $\mu$ is the ...
5
votes
2answers
656 views

Radon-Nikodym derivative as a measurable function in a product space

Let $X$ be a Polish space with the probability measure $P$ and the Borel sigma-algebra. Suppose that $X$ is also a group such that $(x,y)\mapsto xy^{-1}$ is Borel measurable and the probability $P$ is ...
1
vote
1answer
81 views

Feller continuity of the stochastic kernel: compact set

This question is an extension of Feller continuity of the stochastic kernel. Nate Eldredge provided a nice counterexample, but I failed trying to extend it to the compact set $B$. The setting is the ...
4
votes
1answer
294 views

Feller continuity of the stochastic kernel

Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
5
votes
4answers
453 views

Integrating $f(x)=x$ for $x \in C$, the Cantor set, with respect to a certain measure

From general measure theory, I think it's possible to create a measure space $(C, \mathcal{M_\phi}, m_\phi)$ where $C$ is the middle third Cantor set. Now the measure $m_\phi$ is defined as ...
8
votes
1answer
811 views

Integral of differential form and integral of measure

I am trying to understand the relations and differences between integral of differential form and integral of measure. From Wikipedia: On a general differentiable manifold (without additional ...
25
votes
4answers
4k views

Construction of a Borel set with positive but not full measure in each interval

I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere. To be precise, if $\mu$ denotes Lebesgue ...
2
votes
2answers
980 views

Outer measure (Lebesgue outer measure)

Let $A,B \subset \mathbb{R}$ and $m^*(A)=m^*(B)=1$ and $m^*(A\cup B)=2$. Prove that $m^*(A\cap B)=0$. I tried every way I can think of but I do not know how to figure this out. Only properties that ...
5
votes
1answer
190 views

Convergence of a double sum

Let $(a_i)_{i=1}^\infty$ be a sequence of positive numbers such that $\sum_1^\infty a_i < \infty$. What can we say about the double series $$\sum_{i, j=1}^\infty a_{i+ j}^p\ ?$$ Can we find ...
3
votes
0answers
125 views

Semi-partition or pre-partition

For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$. Does anybody met the name for a ...
2
votes
0answers
144 views

Family of measure that admits a continuous density

This question is a generalization of an example provided in Absolute continuous family of measures. Consider a metric space $(X,\rho)$ with a Borel $\sigma$-algebra $\mathcal B(X)$. Consider a ...
2
votes
1answer
199 views

Convergence of inner product using Cauchy-Schwarz

I'm reading a paper in which the following argument is made (in the proof of Theorem 7). I will try to provide just the essentials necessary to ask my question, in particular omitting the ...
3
votes
1answer
132 views

Absolute continuous family of measures

Consider the following family of measures on $(\mathbb R,\mathcal B(\mathbb R))$: $$ K_x(A) = \begin{cases} \int\limits_A \frac{1}{|x|\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy&,\text{ if }x\neq 0, ...
1
vote
0answers
200 views

Tight convergence

Let $(M,g)$ be a closed, Riemannian manifold. Let $u_n$ be a sequence of smooth, positive functions on $M$ such that the $L^1(M)$ norm of the sequence is bounded uniformly. Can we say that $u_n$ ...
2
votes
1answer
129 views

A sequence of functions $f_n \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$

Consider a sequence of functions $\{f_n \}\in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ , convergent to $f$ in $L^1(\mathbb{R})$ and to $g$ in $L^2(\mathbb{R})$. Prove that $f=g$ a.e. What I understood ...
5
votes
3answers
175 views

A problem about the limit of an integral

Let $ g(x) $ be a continuous periodic function of period 1 on $\mathbb{R}$. Prove that for any integrable function $f(x)$ on $[0,1]$, $$ \lim_{n \to \infty}\int_0^{1}f(x)g(nx)dx= ...
2
votes
1answer
438 views

Decomposition of a finite measure on the sum of an atomless measure and a purely atomic measure

Let $\mu$ be a positive finite measure in some $\sigma$-field $\cal{S}$ of subsets of $X$. How find a decomposition $X=B \cup C$ on the disjoint sum of two measurable sets $B, C$ such that the ...
1
vote
1answer
878 views

a nonmeasurable set $E$ of finite measure and a $G_{\delta}$ set $G$ that contains $E$

I understand that the measurability of a set is equivalent for the existence of a $G_{\delta}$ set $G$ that contains the set and has the same outer measure. However, I do not know how to answer this ...
2
votes
1answer
143 views

Measure of $R_d \times \{0\}$

Let $X=R_d \times R$, where $R_d$ denotes the set of real numbers with the discrete topology and $R$ the set of real numbers with the natural topology. For every $f \in C_c(X)$, one has $f(\{x\} ...
1
vote
1answer
111 views

Measure in the Riesz representation theorem for closed subsets

Is it true that in the Riesz representation theorem $\mu(F)=\sup\{\Lambda(f): f\in C_c, 0\leq f \leq 1, \operatorname{supp} f \subset F \}$ for every compact (or closed) subset $F$? (It is known ...
2
votes
0answers
410 views

Is the measurability of the set E required for this problem to be right or have a solution? [closed]

This is one problem from my text book and since this book is new edition, I have been finding many typos or errors in this book. So I am not sure if this problem has an error that it should have ...
2
votes
1answer
1k views

Product of two Lebesgue integrable functions

In general, I know that it is not necessarily the case that the product of two Lebesgue integrable functions $f,g$ will be Lebesgue integrable. But I was reading in a textbook that if at least one of ...
3
votes
1answer
159 views

Measure in the Riesz representation theorem on open subsets

Let $X$ be locally compact Hausdorff space and $\Lambda$ be a positive linear functional on $C_c(X)$. It is known [W.Rudin, Real and complex analysis, th.2.14] that the measure $\mu$ in the Riesz ...
4
votes
1answer
133 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
154 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
195 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
243 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
187 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
249 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
152 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 $\lim_{x \to \infty} f(x) = 0$ .
52
votes
3answers
2k 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
194 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
689 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
218 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
265 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
330 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 ...
16
votes
3answers
2k 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
606 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
467 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
195 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
159 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
142 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
278 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 ...
26
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 ...
6
votes
2answers
298 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
248 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 ...
5
votes
2answers
2k 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?
3
votes
1answer
284 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 ...