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

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58 views

square integrable and compactness

I was wondering if it is possible to define a n-dimensional, square integrable, stochastic process taking values in a compact subset of $\mathbb{R}^{n}$.
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75 views

How to prove this problem about integrable function?

If an integrable function $f(x)\ge0$ a.e., then $\int fd\mu\ge0$. Any hint is appreciated.
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1answer
65 views

If f is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function g does that imply that f=0 a.e

If $f$ is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function $g$ does that imply that $f=0$ a.e
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1answer
63 views

How can I show this inequality?

Let $\lambda B=\{x\in\mathbb{R}^n:\ \|x\|<\lambda\}$. Let $\eta>0$, $r_n\in (0,\eta)$ and $r_n\rightarrow \eta$. Suppose $u$ is a measurable function defined in $\eta B$. How can i show that ...
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364 views

Real valued function of two variables defined on a square with area one, Partial derivatives exist and bounded by an Lebesque intergrable function

This problem in my real analysis textbook has been, let's just say, troubling me. Here is the problem: Let $f$ be a real-valued function of two variables $(x,y)$ that is defined on the square ...
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1answer
70 views

Poisson distribution and probability distributions

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
85 views

Uniform integrability to use DCT

Suppose I have a family $F:=\{f_\alpha\}$, $\alpha \in J$ (index set) of positive functions, a function $L$ increasing, with values in $\mathbb{R}$ such that $L^+(F):=\{L^+(f_\alpha);\alpha \in J\}$ ...
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1answer
122 views

measure theory -Lebesgue measure problem 1

Please help me to do the following problem... For $k>0$ and $A$ is a subset of $\mathbb R$,let $kA=\{kx:x∈A\}$ Show that $m^{*}(kA)=k m^{*}(A)$ $A$ is measurable if and only if $kA$ is ...
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36 views

What is the measure of $(x:f(x)<2^k)$ when $f(x) = |x|^{-a}$? in $R^d$?

I'm confused about measure in $R^d$. What is the measure of $E_k=\{x:f(x)>2^k\}$ when $f(x) = |x|^{-a}, |x|\leq1$, and $f(x)=0, |x|>1$, in $R^d$? My thought is that for $k\leq0$ the set ...
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1answer
210 views

absolutely continuous and g satisfies a Lipschitz condition

If $f$ is absolutely continuous and $g$ satisfies a Lipschitz condition, then $g \circ f$ is absolutely continuous. If $f$ is absolutely continuous and strictly increasing and $g$ is absolutely ...
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34 views

Does $F-F=[0,1] \mod 1$ imply $\mu F = 1$ for $F$ with positive Lebesgue measure?

This question has come up while playing around with the Steinhaus theorem: Let $F-F$ denote the algebraic difference $\{f-g \mod 1 | f,g \in F\}$. Suppose that $F\subset[0,1]$ with $\mu F>0$ , ...
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1answer
31 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 ...
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2answers
456 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 ...
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1answer
61 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: ...
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1answer
51 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 ...
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1answer
245 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 ...
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1answer
305 views

Lebesgue Integral on a set of measure zero

I need to show that if $f$ is an integrable function on $X$ and $\mu(E)=0 ,\ E\subset X$; then $\int _E f(x) d\mu(x)=0$ . In my attempts I've showed that $\forall \epsilon > 0 \ \ \exists ...
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1answer
75 views

Finding a Dominating Function for a Sequence of Integrable Functions

$\fbox{Hypothesis}$ EDIT: Let $\{t_n\}$ be a sequence of reals s.t. $t_n \rightarrow t_0$ with $t_n \ne t_0$. Let $\{g_{t_n}\}$ be a sequence of integrable functions such that for $n$ large enough, ...
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1answer
139 views

Uniform integrability and Lebesgue convergence

A). Given that $ |X_n| \leq Y $ and $Y \in L$. Try to show $X_n$ is lebesgue integrable. b). Try to give any example for which $X_n \longrightarrow^{L} X$ yet $\not\exists Y \in L$ with $|X_n| \leq ...
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1answer
99 views

Measure Theory Question Involving an $f: \mathbb{R}^2 \rightarrow \mathbb{R}$

I'm having trouble parsing the meaning of the following question and am curious if somebody could confirm that the way I am interpreting the prompt is correct: Let $f(x,t)$ be a real valued ...
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1answer
60 views

Refinement of a finite union of measurable sets.

This is paraphrased from an excerpt on a proof from Stein and Shakarchi's $\textit{Real Analysis}$ on page 52, Suppose we have $\bigcup_{k=1}^N E_k$, where each $E_k$ is measurable and of finite ...
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2answers
429 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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1answer
95 views

Relationship between different$L^p (\Omega, \mathcal{F}, \mu)$ spaces with $\Omega$ uncountable and $\mu$ being a Radon measure

I earlier asked this question but I have not had a general classification in the posted answers there. So here is a new question. I am looking now for some special cases as suggested in one of the ...
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1answer
434 views

Approximating Borel sets by finite unions of intervals

Let $B$ be a bounded Borel set of $\mathbb{R}$, Show that if $A$ is a finite union of disjoint intervals, the Lebesgue measure of $A\triangle B$ can be arbitrarily small. Also show that this remains ...
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1answer
63 views

Problem with convergence of functions

I want to prove that: If f is a continuous function, and ${X_n \to X} $ a.e then ${f(X_n) \to f(X)}$ a.e
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1answer
80 views

Measure theory, integration.

Let $(X,S,\mu)$ be a measure space, and let $f,f_1,f_2,\dots:X\to [0,+\infty]$ be $\mu$-integrable such that $\lim\limits_{n\to\infty}f_n=f$ almost everywhere. Show that: ...
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1answer
122 views

The distribution of the sum of infinite fair coin tosses

This question came up in a course on measure theoretic probability theory. We have had lots of information on the existence of distribution, but no examples of how to find/construct them. Here's the ...
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1answer
313 views

Radon-Nikodym derivative relationship between two Gaussian measures

I am trying to find out the the Radon-Nikodym derivative of $μ_1$ w.r.t $μ_2$ where $μ_1$ is a Gaussian measure on $R^n$ with mean $a$, and standard deviation $σ$, and $μ_2$ is also a Gaussian ...
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1answer
125 views

Elementary question on random variables and Borel measurable mappings

In probability theory we have this definition: DEFINITION: Let $(\Omega, \mathcal{U}, P)$ be a probability space. A mapping $\mathbf{X}: \Omega \to \mathbb{R}^n$ is called an n-dimensional random ...
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1answer
402 views

composition of two measurable function

I've come across this problem in my studies, where the book (Real Analysis (4th Edition) by Royden) gives an example of two measurable functions whose composition is nonmeasurable. The two functions ...
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2answers
848 views

Finitely additive probability measure thats not countably subadditive

How is it that a finitely additive probability measure on a field may not be countably subadditive? I know that the field must be countably additive and thus finite additivity does not suffice, but ...
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1answer
162 views

Surface of spheres with the transformation formula

Let $\omega_n$ the measure of the surface (as obtained with the surface measure) of the unit sphere in $n$ dimensions. E.g. $\omega_2 = 2\pi$. Now let $n\ge 3$. I want to obtain a formula for the ...
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1answer
183 views

Convergence in measure and pointwise convergence in continuity points

Hi can you help me with the following: $\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$. Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous. ...
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1answer
70 views

Finite measure assumption in Klenke's characterization of uniform integrability

I fail to see where the finite measure assumption is used in Klenke's characterization of uniform integrability (the part: Uniform Integrability $\implies$ (ii) in the very last paragraph). As i see ...
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1answer
86 views
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1answer
102 views

Proof of convergence of a sum of mean-consistent estimators

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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1answer
83 views

regularity of Newtonian Potential

I saw in SOLUTION OF POISSON’S EQUATION Let $\eta(t)$ be a smooth function such that $\eta(t)=0$ for $t \le 1, 0 \le \eta \le 1$ everywhere and $\eta(t)=1$ for $t \ge 2$. For $n\ge 3, \Gamma(x) = ...
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1answer
52 views

Double Integral Claim

How can I prove the following inequality: (where $f $ is nice enough) - Given a function $ f(x,y) : \Omega_1 \times \Omega_2 \to \mathbb{R} $ , and $\alpha,C_1,C_2 $ are some constants, ( $\Omega_i$ ...
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1answer
127 views

generating $\sigma$-field of a set

Let $X=(X_t)$ be a stochastic process and we define the raw filtration by $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma (X_s;s\le t)$ Now I want to prove that $\sigma (\mathcal{C})=\mathcal{F}_t$, ...
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1answer
368 views

interchange sum and integral

suppose I have a family of i.i.d standard normal random variables $Y_{n,k}$ and I define $X^N_t:=\sum_{n=0}^N\sum_{k=1}^{2^n}Y_{n,k}\phi_{n,k}(t)$ for $t\in [0,1]$ where $\phi_{n,k}$ are the Schauder ...
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1answer
70 views

Question about an implication in a theorem

There is the following theorem: If $(f_n)$ is a sequence in $L^1$ such that $\sum \|f_n\|_1 < \infty$ then (1) $\sum f_n $ converges almost everywhere (i.e. $\sum f_n(x) = K_x < \infty $) ...
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1answer
137 views

Conditional expectation independence

I'm working on some statistics project and am not getting further because of some stupid prediction that doesn't want to be 0. That's why I was wondering if maybe the following holds: Suppose we have ...
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1answer
106 views

Prove that a product of nullsets in $\mathbb{R}^n$ by a compact set in $\mathbb{R}$ is a nullset in $\mathbb{R}^{n+1}$

Let $K$ be a compact set, $K \subset \mathbb{R}^n \times [a,b]$ and, for each $t \in [a,b]$ define $K_t = \{ x\in \mathbb{R}^n $ ; $(x,t) \in K\}$. If $\forall t \ K_t$ has measure zero in ...
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1answer
186 views

Composition of Measurable and Discontinuous function

Consider a locally-bounded function $f: W \rightarrow X$, $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$. Assume that $f$ is Borel measurable (for every open $O \in \Sigma_X$ ...
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1answer
48 views

approximating $E[f(X)]$ where $f$ is absolutely continuous

Let $X$ be a random variable, is it true that I can find $f_n \in C_c$ such that $E[f_n(X)] \uparrow E[f(X)]$, where $f$ is absolutely continuous and $E[f(X)] < \infty$? I'm a little concerned ...
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1answer
68 views

Cauchy sequence in $\mathcal{L}^p(E)$

Be $1\leq p \leq \infty$. If $(f_{n})_n$ is a cauchy sequence in $\mathcal{L}^p(E)$ that converge pointwise to $f$ almost everywhere , show that $f\in \mathcal{L}^p$ and $\displaystyle \lim_{n \to ...
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1answer
61 views

Show $f(a)=\operatorname{E} \min(aX,Y)$ is a quasilinear function of $a$

Let $X$ and $Y$ be bounded real-valued random variables. Define $$ f(a)=\operatorname{E} \min(aX,Y) $$ Is $f$ a quasilinear function of $a$? That is $f$ is both quasiconvex and quasiconcave. To ...
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1answer
240 views

Measure of image of Lipschitz function is bounded?

I recently watched some measure theory lectures online. They didn't post lecture notes and I can't find which video exactly it was. I think there was a theorem that goes something along the lines of: ...
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1answer
83 views

Question on uniform intergrability

Consider a probability measure $m$ over $W \subseteq{R^m}$, so that $m(W) = 1$. Consider a function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, with compact $X \subset \mathbb{R}^n$, such that ...
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1answer
180 views

Probability space for stochastic processes

In Sinai's book on stochastic processes, the definition for discrete time stochastic processes is "a sequence of random variables $\{X_{n}\}_{n\in{}T}$ defined on a common probability space ...