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

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301 views

The indicator function of an open set is an increasing limit of a sequence of continuous functions

Let $G$ be an open set on some measure space $(\Omega,\Sigma, \mu)$. Show that the indicator function of $G$ is an increasing limit of a sequence of continuous functions. I understand that this is a ...
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177 views

Lebesgue Integral of ${x^2}$ over $[0,1]$

I know its just simple using Riemann Integral. I do not know how can I show the integral of ${x^2}$ over $[0,1]$ equals to $\frac13$ using definition of Lebesgue integral.
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52 views

Upper and Lower Differentiability

Show that if $f$ is defined on $(a, b)$ and $c\in(a, b)$ is a local minimizer for f, then $\underline{D}f(c) \leq 0 \leq \overline{D}f(c)$. Proof: There exists $\delta > 0$ such that $f(c) < ...
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1answer
308 views

Uniform integrability and convergence in mean question

Let the sequence $\left\{ \left| X_n - X \right|^r \right\}$ be uniformly integrable for $r > 0$. This means that $E \left[ \left| X_n - X \right|^r 1 \left\{ \left| X_n - X \right|^r > a ...
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221 views

Poisson distribution and probability of random variables

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
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1answer
197 views

Why is the total variation of a measure finite?

I'm looking at a theorem in my Analysis textbook that says: If $\mu$ is a complex measure on $X$, then $|\mu|(X) < \infty$. I can't seem to get my head around this being true. The following seems ...
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164 views

Equality of two sigma algebras?

Assume that $F_1$ and $F_2$ are two independent sigma fields. We know that union of $F_1$ and $F_2$ is not necessarily a sigma-field. Suppose we define $ \mathcal{A} = \{A \cap B: A\in F_1, B\in F_2\} ...
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48 views

How to understand stationary solution?

How to understand the stationary solution of the stochatic equation: $$X_{n+1}=A_n X_n+B_n$$ And where can I find more information?
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634 views

Convergence in Central Limit Thorem

The convergence in the Central Limit Theorem is weak convergence, which is weaker than convergence in probability. I set it as an exercise to find an example that convergence in distribution does not ...
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1answer
114 views

$\sigma$-algebra on the space of all probabilty measures of a measurable space

I am trying to understand the arguments in a book I am reading. Consider the probability space $\left( X, \mathcal{B} \right)$ and let $\mathcal{P}$ be the set of probability measures on it. Let ...
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50 views

Construction of an integrable function with a function in $L^2$

I have this really simple question, but I cannot figure out the answer. Suppose that $f\in L^2([0,1])$. Is it true that $f/x^5$ will be in $L^1([0,1])$? Thanks! Edit: I was interested in ...
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381 views

About the measurable subsets and the Lipschitz condition

I have, again, a doubt with the measurable subsets. If I have that $T\colon\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is Lipschitz, does $T$ send Lebesgue measurable sets in Lebesgue measurable sets. ...
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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 ...
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422 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 ...
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2answers
191 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 ...
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90 views

Finding a Dominating function for $-x^3 e^{-tx^3}$, $\forall t > 0$ on $\mathbb{R}_{\ge 0}$

Let $f(x,t) = -x^3 e^{-tx^3}$ I'm trying to find a dominating, integrable function over $f$ for all $t \in \mathbb{R}^+$. Specifically, I'm looking for a function $h$ s.t. $\forall t > 0$, we ...
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34 views

How can I get $\int_{\Bbb R^n}e^{itu_1x_1}dP^{X}=\int_{\Bbb R}e^{iu_1x_1}dP^{X_1}$?

Let $(\Omega,\mathcal{A},P)$ be a probability space and $X=(X_1,X_2,\cdots,X_n)$ a random vector on $(\Omega,\mathcal{A},P)$ such that $$ X_k:(\Omega,\mathcal{A})\to({\Bbb R},{\mathcal B}_{\Bbb R}), ...
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147 views

Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
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2answers
87 views

$f_n$ $\in$ $L_2(\mu)$, the limit $ f \in L_2(\mu)$

If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$. But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to ...
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66 views

Inequality between two expectations!

I normally have problem with proving inequalities since there are many different inequalities and I'm usually confused on how to choose a proper one and focus on that to get my problem solved. Here is ...
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141 views

Stochastic integral: $E\left(\int^1_0(W(s))\,ds\int^1_0t(W(t)\right)\,dt$

I need to calculate the expectation of the product between the integral of a Wiener process and the expectation of a Wiener process. Is the same as the expectation of the product between the integral ...
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81 views

Random variables in $\Bbb R^n$

My question is at the bottom, but I'll give the set up first: Let $\mathbb{P}$ be a probability measure on set $\Omega$ equipped with sigma algebra $\mathcal{F}$. A random variable $X:\Omega ...
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691 views

Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$

Let $(X, \mathcal{B}, \mu)$ be a measure space and $f$ a measurable function on $X$ and suppose that $\forall E \in \mathcal{B}$ we have that $\int_E f = 0$. Then I want to show that $f = 0$ almost ...
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111 views

Why should the set have finite measure in the following proposition?

Here is a proposition in Royden: Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise a.e. on $E$ to $f$ and $f$ is finite a.e. on $E$. ...
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75 views

Problem with Sequences in Measures

If ${X_n \to X} $ and ${X_n \to X'}$ in measure, then show that $ \mu(X \neq X') =0 $.
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238 views

Lebesgue measuarable sets under a differentiable bijection

Let $U,V \subseteq \mathbb{R}^{n}$ be open and suppose $A\subseteq U$ are (Lebesgue) measurable. Suppose $\sigma \in C^{1} (U,V)$ be a bijective differentiable function. Then does it follow that ...
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105 views

What can we say about the image of a measureable map in the support of its push forward measure

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $f:\Omega\rightarrow\mathbb{R}^d$ a measurable function. Let $\mu$ be the probability measure defined by $\mu(B):=\mathbb P(f^{-1}(B))$ ...
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139 views

Probability limit of a decreasing (uncountable) family of sets.

Let $t^*\in(0,1)$ and $(\Omega,\mathcal{F},P)$ be a probability space. Suppose I have set $A\in\mathcal{F}$ and an uncountable family of sets $(B_t : t\in[0,1])\subset\mathcal{F}$ with the following ...
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115 views

Fatou's Lemma with $\max$

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$ so that $m(W)=1$. For all $k \in \mathbb{Z}_{\geq 0}$ let $f_k: X \times W \rightarrow \mathbb{R}_{\geq 0}$ be measurable and locally ...
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140 views

$\sigma$-field generated by a r.v.

Let $(\Omega, {\cal F})$ and $(E, {\cal E})$ be measurable spaces and $X$ a r.v. from $\Omega$ into $E$ (i.e., $X$ is ${\cal F}/{\cal E}$-measurable). We assume that $\cal E$ is generated by ${\cal ...
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196 views

$L^p$ Norm continuity boundedness

I'm studying for a Qualifying exam and can't figure out this problem. I see that the limit must be $f(x)$ and can get the boundedness but had trouble with continuity. Any suggestions? Let $f\in ...
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495 views

Is the limit of uniformly integrable functions integrable?

If $\left\{f_n\right\}$ are uniformly integrable and $f_n\overset{a.e.}{\rightarrow}f$ ($f$ measurable), is $f$ integrable? Can "uniformly integrable" be weakened to "integrable"?
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595 views

A property of outer measure for bounded sets of real numbers.

I have a bounded set $E$ of real numbers. I'm in the process of showing that there is a set $G$ that is a countable intersection of open sets $G_i$ such that $E\subseteq G$ and $E,G$ have the same ...
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104 views

Expectation and proofs on $(\Omega,\mathcal{B},P)$ involving moments and MGF

Ok, suppose we have a random variable, X, on $(\Omega,\mathcal{B},P)$ and $r>0$. I am trying to prove the following 4 things: 1- If $E(|X|^r)<\infty$ then $E(|X|^s)<\infty \;\;\;\forall ...
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142 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
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124 views

Variation of sum of measure

A variation of arbitrary complex measure $\nu$ on the measurable set $E$ is called the number $\|\nu\|(E)=\sup \sum_{n=1}^\infty |\nu (E_n)|$, where supremum is taken over all sequences $(E_n)$ such ...
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1k views

When is the Lebesgue integral continuous?

Let $\sigma: [ 0,\infty) \rightarrow [0,\infty]$ be a Borel-measurable, nonnegative, extended-real valued function, when is the Lebesgue integral $\int_{0}^{t}\sigma(u) du$ a continuous function of t? ...
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128 views

Show that $E(|X|)<\infty$ and $E(X_n)\rightarrow E(X)$

After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these ...
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137 views

elementary measure theory problem.

I am trying to show that, a set $E$ in $\left( 0,1\right) $ is such that, if $\left( \alpha,\beta\right) $ is any interval, then $$\mu\left(E \cap \left( \alpha ,\beta \right) \right) \ge \delta ...
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152 views

Partition of a probability measure in a continuous and atomic part

Let $(\mathbb{R}, \mathcal{B}, \mathbb{P})$ be a probability space. I want to show that $\mathbb{P}$ can be written as $\mathbb{P} = \mu + \nu$, where $\mu$ is a continuous measure (no atoms) and ...
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58 views

General nonatomic measure that cannot be expressed as an integral

I read in a paper (Kingman — Poisson Processes, 2005) that: In most cases the mean [of an inhomogenous Poisson process on a set $A$] is given in terms of the rate function $\lambda(x)$ on $S$ by ...
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131 views

Lebesgue measure on $\mathbb{R}/\mathbb{Z}$

I was reading a (brief) introduction about measure theory today and came across the following statement: (Lebesgue measure on $\mathbb{R}/\mathbb{Z}$): There is a unique probability measure $\mu$ ...
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244 views

Convergence in the absence of Dominated Convergence Theorem, and uniform integrability

This question is extended from Resnick's exercise 5.13 in his book A Probability Path. Let the probability space be the Lebesgue interval, that is, $(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and ...
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96 views

Convergence of Expectation for $X_n=\frac nY 1_{\{Y>n\}}$ for any $Y$ such that $P(0\le Y<\infty)=1$

Here is another self-study exercise that I am struggling mightily with: $X_n=\frac nY 1_{\{Y>n\}}$ for any $Y$ such that $P(0\le Y<\infty)=1$ I am told that $X_n\to X$ a.s for some $X$, and am ...
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517 views

$f=0$ almost everywhere

be $G\subseteq \mathbb{R}$ a open set. Be $f:G\rightarrow \mathbb{R}$* measured and for all interval $[a,b] \subseteq G$ to have that $f$ is lebesgue integrable function in $[a,b]$ and $\int_{a}^b f ...
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65 views

$L^p$ measurable functions equality

Suppose $1<p<\infty$ and $f,g \in L^p (X,M,\mu)$. Where $||f||_P$ and $||g||_p$ are non zero, and $||f+g||_p = ||f||_p +||g||_p$ . Proving that equality: $${f \over ||f||_p} = {g \over ||g||_p} ...
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249 views

$\sigma$-field from $(X,Y)$ and Probability measure comparison

Given: $\Omega=\{a,b,c,d\}$ $P_1: a=\frac{1}{6}, b=\frac{1}{3}, c=\frac{1}{3}, d=\frac{1}{6}$ $P_2: a=\frac{1}{3}, b=\frac{1}{6}, c=\frac{1}{6}, d=\frac{1}{3}$ $X(\omega)=2+1_{a,b}(\omega), ...
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114 views

Extra condition (or proof) for $\int{f_{n}}$ converging to a limit, assuming $f_{n}\to f$ pointwise a.e .and $\int{f}<\infty$

I am trying to show given $f_{n}\to f$ pointwise a.e. and $\int{f}<\infty$, it follows the sequence {$\int{f_{n}}$} has a limit. But I am not sure if extra condition is required. Can anyone give me ...
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2answers
865 views

A problem concerning the Measurable function

Suppose that $f\in \mathcal{L}(\mathbb{R})$ and $g\in \mathcal{L}(\mathbb{R})$, $\phi(x,y)=f(y-x)g(x)$, prove that $\phi$ is measurable in $\mathbb{R}^2$. I try to prove this problem by the ...
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2k views

Weighted average vs. weighted mean

Is there a formal difference between weighted average and weighted mean? I get corrected to the latter if I type in the former in wikipedia, and then there is a lot of stuff about the name "average" ...