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

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21 views

If $\int _{-\infty}^{\infty}f=1$ then prove that $\int_{-\infty}^\infty\frac{1}{1+f(x)}=\infty$

Given that $f:\mathbb R\rightarrow (0,\infty)$ is a measurable function. If $\int _{-\infty}^{\infty}f=1$ then prove that $\int_{-\infty}^\infty\dfrac{1}{1+f(x)}=\infty$ Any hints on how to proceed ...
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0answers
10 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a sequence of natural numbers (i.e. this is a probability space of sequence of natural numbers sometimes used in ...
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1answer
20 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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0answers
11 views

Showing properties of sigma algebras and algebras

For the first one, my thought is that $\mathcal{A}$ being finite means it has a finite union which is essentially the same thing as a finite countable union. So the result follows. For the second ...
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0answers
10 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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1answer
5 views

Smallest (sub-) Sigma algebra of a null set

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
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0answers
25 views

Billingsley 2.5a) [on hold]

The field $\mathfrak{F}(\mathcal{D})$ generated by a class $\mathcal{D}$ of subsets of $N$ is defined as the intersection of all fields over $N$ containing $\mathcal{D}$. (a) Show that ...
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1answer
23 views

Limsup, is there an alternative definition or am I missing the spirit of the question?

Let $X$ be the positive integers Let $H$ be $\mathcal{P}(X)$ For finite $E\in H$ $v(E)$ is the number of points in $E$. Define: ...
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1answer
22 views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
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0answers
22 views

Is this an outer measure, if so can someone explain the motivation

I'm studying Measure Theory, and following Halmos's book and measures came first. I like measures, I also picked up a book on Probability, that motivated measures (well specifically probability ...
4
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1answer
43 views

“+”-Sets are measurable.

$A$ is a subset of $\mathbb{R}^2$ that for every $(x,y) \in A$ there is a $\delta >0$ that $(x-\delta , x+\delta) \times \{y\}$ and $\{x\} \times (y-\delta , y+\delta)$ are subsets of $A$. prove ...
1
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2answers
39 views

Measure theory exercise

From measure theory volume 1 by Fremlin, exercise 111Xf: Let $X$ be a set, $\mathcal{A}$ a family of subsets of $X$, and $\Sigma$ the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$. ...
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1answer
12 views

Easy argument in Lemma of Corea formula

I don't understand a presumably easy argument in my textbook. Let $L: \mathbb R^n \to \mathbb R^m$ be a linear map, $n \geq m$, $A \subset \mathbb R^n$ $\lambda^n$-measurable. We assume that $\dim ...
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2answers
14 views

Convergence of a function involving a characteristic function of a decreasing interval

Let $f_k: \mathbb{R} \rightarrow \mathbb{R} , f_k(x) = \frac{1}{\sqrt{x}}\chi_{\left[\frac{1}{2^{k+1}},\frac{1}{2^k}\right]}(x)$ For $k \rightarrow \infty $ the interval ...
1
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1answer
25 views

Algebra vs. Sigma-Algebra Condition

I just wanted to clarify the difference between the Algebra and $\sigma$-algebra: Algebra: If $A_1, A_2 \ldots $ are in $\mathscr A$, then $\bigcup_{i = 1}^{n} A_i \in \mathscr A$ ...
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1answer
22 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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0answers
19 views

Differentiation through the integral sign, more general case

I wondered in which cases, given a measurable space $(A,\mu)$, Banach spaces $E,F$, an open $U\subseteq E$ and $f:A\times U\rightarrow F$, we can conclude that the function $s\mapsto\int_A f(x,y)dx$ ...
0
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0answers
12 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
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1answer
36 views

Showing $\sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty$ implies $f_{n}\rightarrow f$ almost everywhere.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then ...
0
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0answers
32 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
0
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0answers
16 views

estimating a convolution type maximal function

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}_{+}$ be a $C^1$ function with $supp(\phi) \subset B(0,1)$ and $\int \phi = 1$. Define $$\phi_t(x) := t^{-n} \phi({x/t})$$ and set $$ M_{\phi} f(x) := ...
1
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1answer
17 views

Integration with respect to a measure

I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as: $$\int f \ d\mu$$ How does the measure change our old "calculus" notion of ...
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1answer
21 views

Showing that $\mathcal{A}$ being countable $\Rightarrow f(\mathcal{A})$ is countable - (Algebras/Sigma Algebras)

For the first question my idea was to show that $\sigma(f(\mathcal{A})) \subseteq \sigma(\mathcal{A})$ and $\sigma(\mathcal{A}) \subseteq \sigma(f(\mathcal{A}))$. As for the second question I am at ...
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1answer
36 views

Showing that if $A_{1},A_{2},…$ are all algebras then the union of all of them is an algebra [duplicate]

I am not sure how to show this. It seems obvious but maybe its not. The help would be greatly appreciated!
2
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1answer
21 views

Complex Measures: Absolute Continuity [on hold]

Note: This is a lemma for: Spectral Measures: Riemann-Lebesgue Given a positive measure: $$\lambda:\mathcal{A}\to[0,\infty]$$ Consider a complex measure: $$\mu:\mathcal{A}\to\mathbb{C}$$ How to ...
3
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2answers
28 views

existence of a borel probability measure on $[0,1]$ such that $\int f d\mu=\lim_{k\to\infty}\frac {1} {N_k} \sum_{i=1}^{N_k}f(x_i)$ given sequence

Hi I'm really suck with this one, i would really appreciate it if any one can help me with this! prove that for $\{x_i\}\subseteq[0,1]$ there are $1\le N_1<N_2...$ and a probability measure $\mu$ ...
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0answers
14 views

Abstract question in measure theory related to product measure

First of all let me write up some definitions here: (M1),(M2),(M3) are the properties of a measure. (M4) is finite additivity. Hey guys I am having hard time wrapping my head around this question ...
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1answer
15 views

Is this measure finite, $\sigma$-finite, or a probability measure?

I was a little unsure on this problem. I do have some ideas though. The way I thought of translation invariant is that you can take an interval and shift it, and in the process is will still be the ...
0
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0answers
11 views

Showing Brownian motion is measuable

How can I prove Brownian motion is measurable with respect to the corresponding product sigma algebra? I am struggling to extend the measurability from holding for rational times to all times using ...
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3answers
47 views

Union of two $\sigma$-algebras is not $\sigma$-algebra

Here is another very basic analysis problem but that puzzles me: Find an example of set $X$ and its two $\sigma$-algebras $\mathscr A_1$ and $\mathscr A_2$, such that $\mathscr A_1 \cup \mathscr ...
11
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1answer
92 views

Measurability Question?

I assigned the following to a class I'm teaching and, to my embarrassment, I cannot come up with a solution. Let $(X,\mathcal B)$ be a measurable space and let $(f_n)_{n\ge 1}$ be a sequence of ...
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1answer
15 views

A set with non-$\sigma$-algebra monotone class

Working on this very basic analysis problem: Find an example of set $X$ with its monotone class $\mathscr M$ such that $\emptyset, X \in \mathscr M$, but it is not a $\sigma$-algebra. My ...
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0answers
15 views

$C(K)^*$ when K is a countable, compact metric space. [on hold]

If K is a countable, compact metric space, then why $C(K)^*$ consists of only purely atomic measures? Also, why $C(K)^*$ is isometric to $\ell_1$ ? (See Topics in Banach space theory by Albiac and ...
1
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1answer
27 views

Nonmeasureable subset of ${\mathbb{R}}^2$ such that no three points are collinear?

I'm exploring the properties of sets in the plane that do not contain a set of three collinear points. In particular, I'm interested in the "largest" they can be. Things I know so far: Assuming the ...
0
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0answers
25 views

$\sigma$-algebra generated by a topology [on hold]

Suppose that $(X, \mathcal{T})$ is a topological space. My hunch is that the smallest $\sigma$-algebra on $X$ containing $\mathcal{T}$ is the collection of Borel sets obtained from $\mathcal{T}$. Is ...
0
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0answers
27 views

The largest $\sigma$-algebra generated by a subset

It is always possible to find a smallest $\sigma$-algebra that contains any subset $A$ of a given set $X$ (it is, by definition, the intersection of all the $\sigma$-algebras on $X$ that contain $A$). ...
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1answer
26 views

Showing that a collection of intervals (see problem) generates the Borel sigma algebra on $(0,1]$

I would be very appreciative if someone could show me how to do this problem so that I can try to get a better understanding of what a Borel sigma algebra is. Examples are how I learn best so seeing ...
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1answer
22 views

How does a change of measure affect covariance?

Suppose I have the three random variables $X,Y,M$ where $E[M] = 1$ under the measure $P$. Now, suppose I define a new measure $\widetilde P$ so that $\widetilde E[X] = E[M X]$ and $\widetilde E[Y] = ...
0
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1answer
19 views

How would I show that $m(\cup_{n \in \mathbb{N}}A_{n}) = \displaystyle\sum_{n \in \mathbb{N}}m(A_{n})$?

Suppose we have that $A_{1}, A_{2}, A_{3}, ...$ is a countable collection of elements of an algebra $\mathcal{A}$. Suppose also that $m$ is a probability measure on $\mathcal{A}$ where $m(A_{i} \cap ...
3
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1answer
33 views
+50

Prove or disprove $\nu(E)=\lambda(f(E))$ is a measure provided that $f$ is nondecreasing and satisfies the N-condition.

Suppose $f$ is a non-decreasing continuous function from $[a,b]$ to $\mathbb{R}$, and $\lambda$ is the Lebesgue measure in $\mathbb{R^1}$. Also, $f$ satisfies the property that $f$ maps Lebesgue ...
2
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1answer
35 views

$f\in L^p(X,\mu)$ , $f-1\in L^q(X,\mu)$ then $\mu(X) < \infty $

Can some one give a hint how to start to solve : Assume $ 1 \le p,q < \infty $ and $$f\in L^p(X,\mu)$$ now if we assume $$f-1\in L^q(X,\mu)$$ then we have $$\mu (X) < \infty $$ Thanks If ...
2
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0answers
18 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
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2answers
33 views

Does the following set contain the Borel $\sigma$ algebra?

Suppose I have $X = \{(2^{-n-1},2^{-n}]:n \in \mathbb{N} \cup \{0\}\}$ and $ K = \sigma(X)$ where $\sigma(X)$ is a sigma algebra. My question is does $K$ contain the Borel $\sigma$-algebra?
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1answer
9 views

Manipulating the definition of $\sigma$-algebra generated by a family of sets

I manipulated the standard definition of $\sigma$ algebra generated by a family of sets $\mathcal{A}$, $$ \sigma ( \mathcal{A}) := \bigcap \{ \Sigma \ | \ \Sigma \text{ is a $\sigma$-algebra on } X, ...
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0answers
55 views

A Classification problem in measure theory [on hold]

Let $m$ be the Lebesgue measure on $\Bbb R$. I want to find all measurable functions $f:[0,1] \rightarrow [0,1]$ with the following property. $$\forall A\subseteq [0,1]~ (\text{Lebesgue ...
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0answers
7 views

what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
0
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1answer
19 views

Generating Borel algebra - proof

In the second paragraph of the proof it says: "To prove that $\mathcal{B}(\mathbb{R})$ is also generated by the other classes of intervals, it suffices to prove that any interval $]a,b[$ is contained ...
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1answer
44 views

reducible measure functions [on hold]

Let $f:\Bbb R^n \rightarrow \Bbb R^n$ be a nonexpansive map, that is $\|f(x) - f(y)\| \leq \|x-y\|.$ I want to know that if $f$ reduces measure, i.e. if $A,f(A) \subseteq \Bbb R^n $ are measurable, ...
1
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2answers
54 views

Measure Theory Book for My Background / Need

My current Math background is as follows: 1) Read first 7 chapters of Rudin "Principles of Mathematical Analysis" and solved a lot of the given problems. 2) Completed Munkres "Analysis on Manifolds" ...
0
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0answers
19 views

Measure in a locally compact Hausdorff space

Suppose that $X$ is a locally compact Hausdorff space in which every open set is $\sigma$-compact and such that for every compact subset $K$ of $X$ we have $\mu(K) < \infty$. What can be deduced ...