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

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2
votes
1answer
163 views

Independence of two limits

Let $(X_n)$ and $(Y_n)$ be two sequence of random variables. $(X_n)$ and $(Y_n)$ are independent to each other. If $(X_n)$ and $(Y_n)$ have limits in distribution. $(X_n)$ tends to $X$, and $(Y_n)$ ...
1
vote
0answers
47 views

Ergodic transformation [duplicate]

Possible Duplicate: Showing a Transformation increases measure (Ergodic Theory) Hoi, i want to show that the piece-wise linear map $T: [0,1)\to[0,1)$ given by $Tx=3x$ for $x\in [0,1/3)$ ...
2
votes
1answer
285 views

Sigma-algebra from a finite class of sets

This is problem 1.7 from "Measure Theory and Probability Theory" by K. Athreya and S. Lahiri. Let $\Omega$ be a nonempty set and let $\mathcal{B} \equiv \{ B_i: 1 \leq i \leq k < \infty\} \subset ...
2
votes
1answer
600 views

What is a product $\sigma$-algebra?

My question is relatively simple: what is a product $\sigma$-algebra? And why they are important? Can anyone suggest any links of intuitive (possibly with simple figures) explanations? Or, maybe ...
1
vote
1answer
73 views

Minimal conditions for convergence of measurable functions

If $X_i$ is a sequence of $\mathbb{R}^m$-valued random variables that converges either in probability or almost surely to $X$ and if $f$ is some measurable function from $\mathbb{R}^m$ into ...
1
vote
1answer
28 views

Minimizer of $p$-variance

Let $\mu$ be a probability measure on $\Omega$, $X$ a random variable on $\Omega$. It is well known that the quantity $E[(X- c)^2]$is minimized over all $c\in \mathbb R$ by setting $c = E(X)$. What if ...
4
votes
1answer
196 views

How is this book applying Fubini/Tonelli without assuming $\sigma$-finiteness?

I am reading about $L^p$ spaces on this google book and in proposition 1.1.4 (page 4) it writes $$ p\int_0^\infty \alpha^{p-1} \int_X \chi_{\{x:|f(x)|>\alpha\}}d\mu(x)d\alpha = \int_X ...
2
votes
4answers
392 views

Proving that a function is non-measurable.

I need to show that the following real valued function on $\mathbb{R}$ is nonmeasurable: $$ f(x) = \begin{cases} x &\text{$x \in E$} \\ -x &\text{$x \in [0,1] \setminus E$} \end{cases} $$ ...
2
votes
1answer
100 views

Does $\Vert f \Vert_p = \sup_{\Vert g \Vert_q=1}\int fg d\mu$ fail if $f \notin L^p$?

I know that for $p \in [1,\infty]$ if $X$ is $\sigma$-finite (for the $p=\infty$ case) we have $$ \Vert f \Vert_p = \sup_{\substack{g \in L^q\\\Vert g \Vert = 1}} \int_X fg d\mu. $$ I always see it ...
2
votes
1answer
199 views

Finding a linear functional's corresponding measure

I just started looking through the proof of the Riesz Representation Theorem (in Rudin's Real and Complex Analysis), and I am still very confused about several things. I'll just write the statement of ...
2
votes
1answer
483 views

measurable function, measurable set, characteristic function, and simple function

Firstly, Definition 1: function f is measurable if we have a sequence of simple function $s_n$ such that $s_n \to f$. Definition 2: a set $A$ is measurable if characteristic function $\chi_A$ is ...
6
votes
2answers
179 views

A question about measure on $\mathbb{R}^2$

Let $\mu$ be a locally finite Borel measure on $\mathbb{R}^2$, and for every $r\in \mathbb{R}^+$ , $\mu(B(x,2r))<C\mu(B(x,r))$ for some $C\in \mathbb{R}$,where $B(x,r)$ is Euclidean open ball at ...
2
votes
1answer
169 views

A question about an exercise on measure theory

I will write a question from Folland's book. What I want to ask is not the solution of this problem, but the way how to approach it. Question is as follows: If $f \in L^+$ and $\int f < ...
3
votes
2answers
87 views

Proving $\otimes_{i=1}^{i=n}\mathcal{B}_{X_{i}}=\mathcal{B}_{\Pi X_{i}}$

I am given the following exercise: Let $X_{\alpha}$ be a measureable space with $\sigma$-algebra $M_{\alpha}$ , mark $X\triangleq{\displaystyle \prod_{\alpha\in A}X_{\alpha}}$ and ...
5
votes
2answers
112 views

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$?

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$? Think when we look at the equivalence classes of almost ...
2
votes
1answer
46 views

Compact subclasses of $R^\mathbb{N}$

I am following this source: http://www.hss.caltech.edu/~kcb/Notes/Kolmogorov.pdf and agree with everything done in sections 1-3. In section 4, I cannot fill in the detail for for Lemma 4, because I ...
0
votes
1answer
34 views

Proving that if $A$ is countable then $\otimes_{\alpha}M_{\alpha}$ is created by sets of form ${\displaystyle \prod_{\alpha\in A}E_{\alpha}}$

I am given an exercise, the following is the first part of the exercise: let $X_{\alpha}$ be a measureable space with $\sigma-algebra$ $M_{\alpha}$ , mark $$X\triangleq{\displaystyle ...
1
vote
1answer
424 views

Proof of sigma-additivity for measures

I understand the proof for the subadditivity property of the outer measure (using the epsilon/2^n method), but I am not quite clear on the proof for the sigma-additivity property of measures. Most ...
0
votes
1answer
161 views

Property of Borel measure?

If $ \mu $ is a finite Borel measure on $R^n $ and if $B_1(x)$ denotes an open ball of radius 1 centered at x, is it true that for compact subset $K$ of $R^n$ there is a point $x_0$ in $K$ such that ...
3
votes
2answers
408 views

Support of regular Borel Measure

This question is elementary and hence might be a duplicate. From Rudin, Real and Complex Analysis, page 57. Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$: assume $\mu(X)=1$. ...
1
vote
1answer
188 views

essential supremum of a matrix multiplication operator

Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a ...
3
votes
1answer
227 views

Lebesgue Measurable Set

So the question the I am working on is: given $S\subseteq [0,1]$ and that $\lambda^*(S)+\lambda^*(I\setminus S) =1$, show $S$ is $\lambda$-measurable. Where $\lambda^*$ denotes the Lesbegue outer ...
4
votes
1answer
97 views

When does a measure have a density?

Consider a measure space $(X, \Sigma, \mu)$ and another measure $\nu$ on the same space. I'm interested in the conditions under which $\nu$ can be represented by a density function $f$ on $X$, so for ...
2
votes
1answer
598 views

Show that a simple function is measurable if its parts are all measurable

I understand that a simple function $s:\mathbb{R}^2 \to \mathbb{R}$ is any function which assumes only a finite number of distinct values. It can also be written as a linear combination of indicator ...
3
votes
4answers
208 views

Measure and Lebesgue Integral

I got this exercise as homework and I found some problems in solving it. So I hope that someone can help me. Let $f:[0,1] \rightarrow R$ Lebesgue measurable and $S=\{x \in [0,1]:f(x) \in Z\}$. Show ...
1
vote
1answer
962 views

Computing Radon-Nikodym derivative

I learned Radon-Nikodym theorem in class and I know what exactly it is. But I am not sure about how to compute Radon-Nikodym derivative... Any reference does not explicitly say about how to compute ...
4
votes
4answers
604 views

Cardinality of Vitali sets: countably or uncountably infinite?

I am a bit confused about the cardinality of the Vitali sets. Just a quick background on what I gather about their construction so far: We divide the real interval $[0,1]$ into an uncountable number ...
4
votes
1answer
328 views

limsup of intersection of events as a subset of intersection of limsups

Let $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ be two sequences of events in some probability triple $(\Omega, \mathcal{F}, \mathbf{P})$. Now, it is true that $\left(\limsup_n A_n\right) \cap ...
11
votes
4answers
789 views

Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?

I wanna prove that "if $f: \mathbb{R}^n \to \mathbb{R}$ is continuous and satisfies $f=0$ almost everywhere (in the sense of Lebesgue measure), then, $f=0$ everywhere." I am confident that the ...
2
votes
1answer
59 views

measure of the boundary of the support

Let $\mu$ be a Borel probability measure on $\mathbb R^d$. Does the boundary of the support of $\mu$ have measure zero, i.e. do we have $$\mu(\partial(\text{supp}\mu))=0,$$ where we define the support ...
0
votes
2answers
736 views

Limits of infimum and supremum for sequences of functions

I need to show that $-\infty \leq \liminf_{k \to \infty}f_k \leq \limsup_{k \to \infty}f_k \leq \infty$ , where $f_k$ is a sequence of functions from $\mathbb{R}^n$ to $\mathbb{R}$. This seems ...
1
vote
2answers
194 views

Lebesgue measurable sets have the same Lebesgue measure as Borel sets

I have read that if $A$ is Lebesgue-measurable, then there exists Borel sets $B,C$, with $B\subset A\subset C$, such that $m(B) = m(C) = m(A)$. It is clear for me that such a set C exists, just by ...
4
votes
1answer
93 views

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
4
votes
2answers
1k views

$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable

Problem. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a monotone increasing function. Show that $f$ is measurable. Solution. We know that the set of discontinuites of any monotone increasing ...
3
votes
1answer
91 views

Uniform integrablity of measurable functions

How can I show that if family of $f$ is uniformly integrable then so is {$|f|$}? $($by uniformly integrablity: $\forall \epsilon>0 \ \exists \delta>0: |\int_Ef|<\epsilon,\mu(E)<\delta)$ ...
2
votes
1answer
237 views

Limit of a measurable function and the Lebesgue integral

Suppose $\{f_n\}$ is a sequence of lebesgue measurable functions such that $f_n\rightarrow f$, except on a set of measure $0$, as $n\rightarrow\infty$, and $|f_n(x)|\leq g(x)$, where $g$ is ...
4
votes
1answer
205 views

Foundations of measure theory

In measure theory one usually starts with a $\sigma$-algebra $A$ of sets and considers a measure $\mu:A\to [0,\infty]$. I'm interested in abstracting this definition to allow more general domains and ...
2
votes
2answers
121 views

If $A$ and $B$ are separated, is $m^*(A \cup B)=m^*(A)+m^*(B)$?

Suppose that $A,B$ are separated sets of real numbers, that is $$\inf \{|a-b|:a \in A,b \in B\}>0.$$ Is it then true that $$m^*(A \cup B)=m^*(A)+m^*(B),$$ where $m^*$ is the Lebesgue outer ...
3
votes
2answers
376 views

A set in a $\sigma$-algebra that can't be “reached” with countable set-theoretical operations

Can someone please give me an example of a set that lies in a $\sigma$-algebra generated by some set other then the $\sigma$-algebra itself, such that this (the first) set can't be obtained by ...
17
votes
1answer
1k views

Approximating a $\sigma$-algebra by a generating algebra

Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$. Then for all $B\in\cal B$ and ...
6
votes
2answers
1k views

Radon-Nikodym derivative of product measure

For $j=1,2$, let $\nu_{j},\mu_{j}$ be $\sigma$-finite measures on $(X_{j},\mathcal{M}_{j})$ such that $\nu_{j}\ll\mu_{j}$. I want to show that $\nu_{1}\times\nu_{2}\ll\mu_{1}\times\mu_{2}$ and that ...
1
vote
2answers
1k views

Example of strictly subadditive lebesgue outer measure

One of the properties of the Lebesgue outer measure is that it is subadditive and not countably additive. In fact, I have read that even when the sets A_i are disjoint, there is still generally ...
2
votes
1answer
468 views

Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?

Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable? There are a number of equivalent definitions ...
6
votes
1answer
150 views

Measure of a set in $[0,1]$

Let $E \subset [0,1]$ be measurable set. Suppose for each interval $I \subset [0,1]$, $m(E \bigcap I)>1/4 m(I) $. Show that $m(E)=1$. Any hints would be appreciated.
2
votes
2answers
230 views

If integration of arbitrary polynomial with respect to Borel measure $\mu$, over $[0,1]$ vanishes, is it true that $\mu$ equals to $0$ on $ [0,1]$?

I am having difficulties to deal with following problems; Assume $ \displaystyle\int_{[0,1]} x^n d \mu =0$ for all $n$, then is it true that $\mu=0$ on [0,1]? I think it is definitely true.. but I ...
4
votes
2answers
115 views

Limit and Lebesgue integral in a compact

I have problem with the exercise that follows. Let $(z_m)_m \in R^n$ so that $\Vert z_m \Vert \rightarrow \infty$ when $m\to \infty$. Let $f:R^n \rightarrow [-\infty;+\infty]$ integrable. Show ...
0
votes
1answer
64 views

Find $\mu(\partial B_r(0))$ without integral theory

Is it possible to solve the following exercise without any reference to Lebesgue integral? Given $\omega_n:=\mu(B_1(0))$ find $\mu(B_r(0))$ and $\mu (\partial B_r(0))$. First part is very easy: ...
3
votes
1answer
133 views

Convergence of Lebesgue integrable functions in an arbitrary measure.

I'm a bit stuck on this problem, and I was hoping someone could point me in the right direction. Suppose $f, f_1, f_2,\ldots \in L^{1}(\Omega,A,\mu)$ , and further suppose that $\lim_{n \to \infty} ...
3
votes
1answer
120 views

Negative integral on intervals implies negative function?

Let $f \in L^1([0,1])$ be such that for all $t \geq s$, $\displaystyle \int_s^t f(u)du \leq 0$. Is it true that $f\leq 0$ almost everywhere?
0
votes
1answer
60 views

Change of Variables and independent random variables.

Suppose that we have two IID random variables, $X_1, X_2$, carried by a triple $(\Omega,\mathcal{F},P)$. While solving an exercise I ended to a point that I had to see that, $$ \iint\limits_D x_1 ...