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

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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
36 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
428 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
163 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
420 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
191 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
99 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
604 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
209 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
976 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
607 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
329 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
798 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
60 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
744 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
199 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
239 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
206 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
381 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 ...
18
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
472 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
152 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
237 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
116 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
135 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 ...
2
votes
1answer
821 views

Finding Lebesgue Integral of $\frac{1}{\sqrt{x}}$ over $(0,1]$

How do I rigorously discover what $$ \int_{(0,1]} \frac{1}{x^{1/2}} = \underset{0 \le \phi \le \frac{1}{\sqrt{x}}}{\sup} \int_{(0,1]} \phi $$ (for $\phi$ a simple function) is? Note that I have ...
7
votes
1answer
165 views

Given a model of ZF where $ \mathbb{R} $ is the countable union of countable sets, does every subset of $ \mathbb{R} $ have measure zero?

The question basically says it all. It is a well-known result that there exists a model $ \mathcal{M} $ of ZF with the property that $ \mathbb{R}^{\mathcal{M}} $ (here, $ \mathbb{R}^{\mathcal{M}} $ is ...
1
vote
1answer
484 views

Limit of a decreasing sequence of measurable sets.

Let $(X,\mathcal{A})$ be a measurable space, with measure $\mu$. Let $\{E_n\}_{n \in \mathbb{N}} \subseteq \mathcal{A}$ be a sequence of measurable sets, with $E_{n+1} \subseteq E_n, \ \forall n \in ...
0
votes
2answers
138 views

Can we use Fubini's Theorem?

Is there any special technique to deal with the distribution of sum of two random variables where they are not independent? For example I have concluded that if $X =_p W$ and $Y=_pZ$ ($=_p$ means ...
6
votes
1answer
138 views

measure theory question about sum of sequence of functions

Let $(f_n)_{n\ge1}$ be a sequence of measurable real valued functions. Prove that there exist a sequence of constants $c_n$ $>0$ such that $\sum_{i=1}^{\infty} c_nf_n $ converges for almost every x ...
5
votes
1answer
47 views

An integral with respect to a coupling for a Markov chain

Let $(X,\mathcal{B}(X))$ be a measurable space with Borel algebra and $P\colon X\times \mathcal{B}(X) \rightarrow \left[0,1 \right]$ a stochastic kernel. We assume that $X$ is a separable metric ...
1
vote
1answer
125 views

Show a function is in $L_\infty$

Let's assume we're working on a measure space $(X,\Sigma,\mu)$, where $\mu$ is a $\sigma$-finite measure. Suppose that $g$ is a measurable function such that $\forall f\in L^2$, $||fg||_2\leq ...
3
votes
1answer
145 views

Marginals and compactness in the narrow topology

I've read in a working paper (bottom of page 9) that the following is a "standard result": Let $A$ be a compact metric space and $T$ be a Polish space. Let $\rho$ be a Borel probability ...
4
votes
1answer
97 views

Two measures on a measurable space

I want to find an example of two finite measures $\mu$ and $\nu$ on a measure space $(X,S)$ with $\mu(X)=\nu(X)$ such that {$A\in S: \mu(A)=\nu(A)$} is not a $\sigma$-algebra. Can someone help me?
1
vote
1answer
102 views

How do we know that this set is measurable?

Let $(\Omega, \mathbb{A},P)$ be a probability triple and $X_n$ be a sequence of random variables. $X_n$ converges almost surely if and only if $Prob(w \in \Omega:X_n(w) \to X(w) \ as \ n \ \to \ ...
5
votes
3answers
3k views

Examples of non-measurable sets in $\mathbb{R}$

I'm a newcomer in real analysis. I am leaning the concept of measurable by myself using Royden's book "Real Analysis". I have a question regarding measurable sets. The following definition comes from ...
1
vote
2answers
356 views

Proof that a set is a sigma algebra

Let $(\Omega, \mathcal{A})$ be a measure space and $A\subset \mathcal{A}$. Show that $\sigma(A) = \sigma(\sigma(A))$. Do we have to use that the minimal sigma algebra is the intersection of all sigma ...
1
vote
2answers
289 views

Points of a Measure Zero Sets Covered by Intervals Infinitely Many Times

Given a measure zero set $E$, by definition we have forall $\varepsilon > 0$, a covering of $E$ by intervals whose lengths sum to $< \varepsilon$. I want to prove that we can cover $E$ in ...