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

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Measure on Product Set

Consider a finite sequence of $\sigma$-finite measure spaces $(\Omega_i, \mathcal{F}_i, \mu_i)$. Constructing the product measurable space $$ (\Omega_1 \times \cdots \times \Omega_n, \mathcal{F}_1 ...
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0answers
12 views

Dimension of a measure in terms of linear form on continuous functions

So this might be a bit of a weird question, but here goes. It is well-known (Riesz representation theorem) that the dual space of continuous functions on a compact $K$ identifies with the space of ...
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1answer
21 views

basic notions of measure theory: differences?

Could you help me differentiating the following notions of measure theory: law, probability, probability density, probability measure, probability distribution, distribution, distribution function. ...
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1answer
34 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
3
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1answer
32 views

Joint Distribution Implies Independence…?

Consider the measurable space $(\mathbb{R}, \mathcal{B})$ and a probability space $(\Omega, \mathcal{F}, P)$. Define a finite sequence of random variables $X_1,\ldots,X_n: \Omega \to \mathbb{R}$. ...
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1answer
12 views

Equivalence of definitions of Gaussian Measure

Wikipedia's article on Gaussian measures notes this as the definition of Gaussian measures: $\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 ...
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1answer
58 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
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0answers
7 views

Sunrise Lemma, left and right maximal functions

Would anyone be able to provide a proof of the following lemma: Where $| \cdot |$ is the Lebesque measure and $M_L$ and $M_R$ are the left and right maximal functions defined on $\mathbb{R}$ ...
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10 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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0answers
16 views

Lebesgue integral of a ratio of Lebesgue densities

I need a hint to solve the following problem: $P$ is a probability mass on $\mathcal B(\mathbb R)$ with a Lebesgue density $h$, $f$ is another Lebesgue density. I need to show that $\int ...
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1answer
22 views

How does the Lebesgue measure measure non-cartesian product sets?

Consider the measure space $(\mathbb{R} \times \mathbb{R}, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda)$ where $\lambda$ is the Lebesgue measure on $(\mathbb{R}, \mathcal{B})$. Since ...
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1answer
28 views

$E(f(|X_n|))$ property implies uniform integrability?

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
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0answers
33 views

Is a limit of measure of a sequence of sets equal to measure of limit of the sequence of sets?

I'm sitting at the same question desk as this: Limit of the measure of the converging sequence of sets. Actually, I can't prove it neither. PA6OTA gave a hint to show there is subsequence $A_{n_k}$ ...
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1answer
21 views

Normally Distributed = Absolute Continuity?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. A random variable $X: \Omega \to \mathbb{R}$ is said to have the standard normal distribution if it has the density $f:\mathbb{R} \to ...
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0answers
32 views

Product Topology and Borel-$\sigma$-algebra

Let $S=\left\{1,2,...,n\right\}$ be equipped with discrete topology and let $X=S^{\mathbb{Z}}$. Then the so-called cylinder sets $$ [s_0,s_1,...,s_m]_n:=\left\{x\in X: \forall 0\leqslant i\leqslant ...
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1answer
43 views

Union of a null set and a non-measurable set

Suppose $S$ is a non-measurable set (wrt the Lebesgue measure) and $N$ has measure 0. What can be said about $S\cup N$? Is it also non-measurable?
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2answers
66 views

What is the difference(s) between $(a,\infty)$ and $(a,\infty]$?

I am studying H. L. Royden's Real Analysis which includes some introduction to Measure Theory; and I encountered $(a,\infty]$ instead of $(a,\infty)$ for the first time! What is the difference(s) ...
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1answer
40 views

A problem on measure restriction

Definition of measurable space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of measure: Let $(\Omega, F)$ be a ...
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104 views

Find measure such that…

I've a very concrete problem I can't solve. Consider the following function $k: [0,1]^2 \to \mathbb{R}:$ $$ k(x,y)=\begin{cases} 1 &\text{if } y > x \\ -1 &\text{if } x- \frac{1}{2} < ...
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1answer
21 views

Proof of “triangle inequality” for outer measures: $\left|\mu^*(A) - \mu^*(B)\right| \le \mu^*(A \Delta B)$

We use the following notation. $X$ is a set, $\Omega_0$ is a ring of subsets of $X$, $\mu_0$ is a premeasure on $\Omega_0$, and $\mu^*$ is the associated outer measure, defined for subsets $A$ of $X$ ...
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18 views

Product Sigma Algebra, Explicitly

Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be measurable spaces and define the product sigma algebra on $\Omega_1 \times \Omega_2$ as $$ \mathcal{F}_1 \otimes \mathcal{F}_2 = ...
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5answers
112 views

What we get if we add 1/2 infinite times [on hold]

I want to know if this is correct We have this sums: $$S1=1-1+1-1+1-1+1-1+1-1...=\frac12$$ $$S2=1-2+3-4+5-6+7-8...=\frac14$$ $$S3=1+2+3+4+5+6+7+8...=-\frac{1}{12}$$ If we take ...
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1answer
24 views

If ${f_n}$ converges to $f$ in $L_p$ sense and to $f'$ point-wisely, does it mean $f=f' a.e.$?

The question came into my mind when I read a theorem from Kubrusly's "Measure Theory: a First Course", saying that if $f_n\rightarrow f'$ uniformly and $f_n\rightarrow f''$ in $L_p$ sense, then ...
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28 views

Why is this definition of the mesaure $\mu$ not properly?

Consider the measurable space $$ (\left\{a,b,c\right\}^{\mathbb{Z}},\mathfrak{A}), $$ where $\mathfrak{A}$ is the product topology on $\left\{a,b,c\right\}^{\mathbb{Z}}$ which is generated by the ...
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1answer
37 views

What is Skorohod's Represntation Theorem Saying?

From Wikipedia: Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. ...
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1answer
19 views

Series Test for Integrability via the Distribition Function

I imagine that the following question has a well known (and perhaps, easily obtainable) answer, but I can't find it by myself nor along the references that I have in mind so far. So, if $f$ is a ...
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1answer
16 views

Property of complex borel measures with absolutely convergent Fourier series (Wiener algebra)

Let $\mu$ be a complex Borel measure on the circle $\mathbb{R}/\mathbb{Z}$ with $$ \sum_{n \in \mathbb{Z}} \lvert\hat{\mu}(n)\rvert < \infty. $$ How does it follow that $d\mu(x) = f(x) dx$ for $f$ ...
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0answers
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Question about the measure induced from another measure, problem 1.22 from Folland

I'm self-learning Measure Theory using Real Analysis book of Folland. Unfortunately, I got stuck in this problem and couldn't find any clue to solve this. Can someone help me, or give me some hint so ...
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Theorem of Portmanteau: It suffices to show it for a base?

I have a question to the Theorem of Portmenteau, see here. Two equivalent statements to $P_n\to P$ weakly, are (1) $\limsup_n P_n(C)\leq P(C)$ for all closed sets $C$. (2) $\liminf_n P_n(O)\geq ...
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0answers
15 views

Proof for factorization lemma

I am confused by the proof of Factorization lemma. Specifically the part "f is a step function" confuses me. $f$ has a representation where all $A_i$ are disjunctive. I prefer to work with this ...
4
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0answers
48 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
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2answers
63 views

Lebesgue integrable function over $(0,1)$ vs $[0,1]$

Up till now, I thought saying $u \in L^2([0,1])$ is the same as saying $u \in L^2((0,1))$, because I see people emphasizing "$u$ is Lebesgue integrable over $[0,1)$". I thought the whole point of the ...
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0answers
24 views

Approximation Lemma for Riemann-integrable functions

In the following let $f : [a,b] \to \mathbb R$ be bounded functions. For a regulated function, the integral could be written as the limit $$ \int_a^b f(x) dx = \lim_{n\to \infty} \int_a^b ...
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2answers
38 views

Bounded function on compact interval that is not Lebesgue integrable

Is there an example of a bounded function $f : [a,b] \to \mathbb R$ which is not Lebesgue integrable?
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1answer
77 views

Lebesgue dominated convergence problem

This is an old exam problem: Evaluate $$\large{\lim_{n\rightarrow\infty}\int_0^{\frac{n\pi}{2}}\frac{1}{n}e^{-x}\tan{\frac{x}{n}}\,\text{d}x.}$$ My idea is to let ...
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0answers
18 views

A problem about a family of mesurable fuctions

Let $(X,\mathcal{M},\mu)$ a measure set such that $\mu(X)<\infty$ and $\mathcal{F}$ a family of $\mu$-measurable functions. Let $E(f,t)=\{x\in X\mid f(x)\geq t\}$ with $f\in\mathcal{F}$. If ...
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1answer
46 views

Borel Measures: Lusin

I'm trying to self-learn. Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\mu\geq0$$ Regard a measurable: ...
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1answer
26 views

Interplay of Hausdorff metric and Lebesgue measure

Consider the space $\mathcal{K}(\mathbb{R}^{n})$ of compact subsets of $\mathbb{R}^{n}$ endowed with the Hausdorff metric $\rho$, and let $\lambda$ denote the $n$-dimensional Lebesgue measure on ...
2
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1answer
37 views

Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [on hold]

I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus. Actually I am unable to get direction ...
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1answer
35 views

Integral of a nonnegative Lebesgue-measurable function on $ [0,1] $.

Let $ f $ be a nonnegative Lebesgue-measurable function on $ [0,1] $. Suppose that $ f $ is bounded above by $ 1 $ and that $ \displaystyle \int_{[0,1]} f = 1 $. Problem. Show that $ f(x) = 1 $ ...
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48 views

A sequence of positive finite measures has a positive measure in the weak limit.

I think the statement in the header is true, but I haven't been able to find a proof for it. Consider the measurable space $(\mathbb{R},\mathcal{B})$, i.e. the real number line with the Borel ...
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1answer
13 views

Definition of lebesgue integral with respect to measure $\mu$ [duplicate]

In Rudin's Real and Complex Analysis, the Lebesgue integral is defined as: L et $(X,m,\mu)$ be a measure space, where $X$ is a set, $m$ is a $\sigma$ algebra on $X$ and $\mu$ is a measure. Then, if ...
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0answers
31 views

Breaking probability measure into conditional probability measure and marginal probability measure.

I would like to know under what conditions on $\rho$ - borel probability measure on $ X \times Y$ - the following statement holds true $$ \int_{X \times Y} \varphi(x,y) \, d\rho = \int_X \left( ...
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1answer
31 views

Combining Fubini and Tonelli's in one single Assumption

I am referring to the statements on Wikipedia, there it is said that Fubini's Theorem states that if $f : X\times Y \to \mathbb R$ is integrable, then $$ \int_X \left( \int_Y f(x,y) dy\right) dx = ...
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1answer
21 views

Partial Integration for measures

I have the following formula in mind, $\mu$ a measure on $\mathbb{R}$. Any sigma-finite measure on $\mathbb{R}$ can be decomposed into a absolut continuous part, a "point measure" and a singular ...
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26 views

Approximation by measure with finite support

Can a Borel probability measure on a Polish space be arbitrarily approximated in the total variation metric by a probability measure with finite support?
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Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

Problem Suppose $f$ is a non-negative integrable function on a measure space $(X,\mathcal{A},\mu).$ Prove that $$\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$$ Attempt Let $E_t=\{x:f(x)\geq t\}$ ...
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23 views

measure-preserving transformations are spectrally isomorphic

If $(X_{1}, \mathcal{B}_{1}, m_{1})$ and $(X_{2}, \mathcal{B}_{2}, m_{2})$ are probability spaces together with measure-preserving transformations $T_{1}:X_{1}\to X_{1}$,$T_{2}:X_{2}\to X_{2}$. How ...
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1answer
14 views

If$\mu$ is $\sigma$- finite, $\epsilon>0$, there exists $A\in \mathcal{A}$ such that $\mu(A)<\infty$ and $\epsilon+\int_A f>\int f$

Problem Let $X\mathcal{A},\mu$ be a $\sigma$-finite measure space. Suppose $f$ is non-negative and integrable. Prove that if $\epsilon>0$, there exists $A\in \mathcal{A}$ such that ...
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1answer
26 views

Borel Measures: Coproduct

I need this thread as lemma! (See the advice: SE: Q&A) Given Borel spaces $\Omega_\lambda$. Consider the coproduct: ...