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

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6 views

General set in product space approximated by rectangle sets

Let $(E^k,\mathcal{E}^k,\mu^k)$ be a product measure space. By a rectangle set in $E^k$, we mean a set of the form $A_1\times\ldots\times A_k$ where each $A_i\in \mathcal{E}$. My question is, for ...
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1answer
44 views

Question about Vitali Covering (from a Lemma in Royden and Fitzpatrick's book)

Definition. For a real valued function $f$ and an interior point $x$ of its domain, the uppper derivative of $f$ at $x$ denoted by $\overline{D}f(x)$ is defined as follows: ...
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0answers
26 views

help with a proof on Doob's Submartingale inequality - application of chebychev's inequality

I am stuck on a final step of the proof, we have that $(X_n)$ are non negative submartingale, and $c>0$. We let $T = \inf \{n: X_n > c \} \wedge N$ which is a stopping time. Let $E \{ ...
2
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30 views

Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
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19 views

Quotient of measurable functions

I need to show that if $f,g : X \to \mathbb{R}$ are measurables with respect to the $\sigma- algebra$ $S$ of $X$, and $g(x) \neq 0, \forall x \in X$, then $f/g: X \to \mathbb{R}$ is measurable. So ...
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0answers
24 views

Weak convergence in probability and functional analysis

Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have ...
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2answers
47 views

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function that is in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$. Here's what I have so far. $f\in L^2 ...
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24 views

Show $A$ is an algebra of sets of $X$ [duplicate]

I'm having trouble with this problem, and not sure where to start with. I'm not sure if this is related to Borel algebra since they have very similar construction. Could someone help me with the ...
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1answer
11 views

Functions h and r: prove $||h||_p = sup_{||r||_q \le 1} \int_E{rh}$

Functions h and r: prove $||h||_p = sup_{||r||_q \le 1} \int_E{rh}$, where $1/p + 1/q =1$. Using Holder's Inequality, I can prove that $||h||_p \le sup_{||r||_q \le 1} \int_E{rh}$, but I'm having ...
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22 views

Can limit be taken inside the measure?

If $\lim_{n\rightarrow \infty }m(E_n) = 0$ is given, then can we take the limit inside the function? For example, in this case can I conclude $m(\lim_{n\rightarrow \infty } E_n) = 0$?
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0answers
17 views

Existence of a locally essentially unbounded integrable function

Does there exist an integrable function $f\colon [0,1]\to \mathbb{R}_+$ such that for every $0\leq a < b\leq 1$ we have $\| \chi_{(a,b)} f\|_\infty = + \infty$?
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1answer
43 views

Almost everywhere convergence of random variables

This is a question that my teacher is having us do for homework but I think there might be a typo in it. I was hoping if someone clear this up for me. The sequence $\{X_n\}$ of random variables ...
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1answer
26 views

A Property of Martingale of Sum of i.i.d. Random Variables

I am trying to solve the following problem: Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with finite mean. Let $F_n =\sigma(Y_1,...,Y_n)$. Let $\tau$ be a stopping time ...
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19 views

What are the two levels of probability theory?

What I mean is what is the Probability theory of using integrals called? (typically undergraduate course, Probability Theory I) Then what is the probability using measure theory? (typically graduate ...
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1answer
27 views

Measurability of a set constructed from another measurable set

Let $E$ be a measurable subset of the real numbers. I define $\sigma(E) = \{ (x,y) \in \mathbb{R}^2 | x-y \in E\}$. I would like to prove that this set $\sigma(E)$ is Lebesgue measurable in ...
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28 views

Measurability of integrals with respect to different measures [closed]

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
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1answer
25 views

Let $\nu$ be a signed measure, then E is $\nu$-null iff $|\nu|(E)=0$

I am having trouble with understanding the proof of the forward direction. In all the proofs I have seen it's always the case that the following fact is implied. Taking a Hahn Decomposition $X=P\cup ...
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0answers
29 views

Interesting question on signed measures

I have managed to do the first part, but I am stuck on the second. I compelted the first part by considering $ A \cap (E_1 \backslash E_2) \subset E_1$ and $A \cap (E_1 \backslash E_2) \subset ...
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2answers
55 views

What is the advantage of Borel sigma algebras in defining probability spaces?

I'm trying to get the central concepts correct, so I'm going to express them without embellishment. A Borel $\sigma$ algebra is defined as a sigma algebra generated by a topological space ...
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2answers
24 views

Measurable Functions over finite or countables sets

Let $X$ be a countable or finite set and $u$ a measure over $(X,\Sigma)$. Let $A=${$A_1,A_2,...$} be the set of atoms of $(X,\Sigma)$. Prove that if $f: X\to \mathbb R $ is a measurable function, then ...
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1answer
33 views

half-closed intervals and Lebesgue measures

I am reading Bartle's book. define $$K=\{ a \in \mathbb{Q}\,|\, 0 < a \le 1\}$$ and define $A$ by the family of all finite unions of half-closed intervals in the form of $$\{a \in K\, |\, x ...
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12 views

Show that measurable sets of finite measure satisfy this property

If $E \subset \mathcal{R}^2$ has finite measure and satisfies $m(E \cap (E+t)) \to 0$ as $t \to 0$ prove that $m(E)=0$. If $E$ has finite measure then we approximate it with a finite collection of ...
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12 views

Spectral measure of a stationary time series

Let $(Z_t)$ be white noise with $E[Z_t^2]=1$ and $A$ and $B$ random variables such that $E[A] = E[B] = 0$, $E[A^2] = E[B^2] = 1$, $A$, $B$ and the infinite sequence $(Z_t)$ are independent ($(Z_t)$ ...
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3answers
43 views

Show that continuous functions on $[0,1]$ satisfy this property

If $f \in C[0,1]$ prove that $$ \lim_{n \to \infty} n\int_0^1e^{-nx}f(x)dx $$ exists and find the limit. I can show that $|g_n|$ are bounded by $M=\max(f)$. After some test functions I suspect that ...
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11 views

How to prove the independent of these 2 $\sigma$-algebras?

Let $(\Omega,F,\mathbb{P})$ be a probability space. Let $(F_i)_{i\in\mathbb{N}^*}$ be a sequence of mutually independent sub-$\sigma$-algebras of $F$. I want to prove that: $$A=\sigma(\bigcup_{m\leq ...
3
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3answers
48 views

Prove $b-a \le \sum^n_{i=1}(b_i-a_i)$ by induction

Show that if the closed interval $[a,b]$ is covered by finitely many open intervals $(a_1,b_1), ...,(a_n,b_n)$, then $$b-a \le \sum^n_{i=1}(b_i-a_i)$$. I know that $(a_1,b_1), ...,(a_n,b_n)$ form an ...
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3answers
41 views

Prove that half-open sets in $\mathbb{R}$ are measurable

Self-learning these concepts, so please be tolerant with imprecise terminology... Defining the standard topology on the real line $\mathbb {R}$ as all the open intervals, a Borel $\sigma$-algebra is ...
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2answers
36 views

Lebesgue Integrability of $\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$

Given $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(0)=0$ and $f(x)=\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$ for $x\in \mathbb{R}-\{0\}$, can someone please give me a rigorous proof ...
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0answers
13 views

How to recover a measure from the product

Let $(X,\mathcal{F}_X), (Y,\mathcal{F}_Y)$ be measurable spaces and $\mu :\mathcal{F}\rightarrow [0,\infty]$ be a measure (assume that $\mathcal{F}\supseteq \mathcal{F}_X\otimes\mathcal{F}_Y$). I do ...
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1answer
39 views

$f\in L^{1}[0,1]$ Show $\lim_{n\to\infty}\int_{0}^{1}|f(x)|^{\frac{1}{n}}dx = m(\left\{ {x:f(x)\neq 0}\right\} )$

The following is from a Sample Exam question I am studying from, and the question has stumped me. $$f\in L^{1}[0,1]$$ $$\lim_{n\to\infty}\int_{0}^{1}|f(x)|^{\frac{1}{n}}dx = m(\left\{ {x:f(x)\neq ...
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1answer
20 views

If $A \cup B$ is measurable and $m(A \cup B) = m^*(A) + m^*(B) < \infty$ then A and B are measurables.

That's it. I've only been able to find that, since $A \cup B$ is measurable: $$m^*(A) + m^*(B) = m(A \cup B) = m_*(A \cup B) \geqslant m_*(A) + m_*(B)$$ Maybe using too that if C is measurable and D ...
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18 views

Measure of $|g| = ||g||_\infty$, with $g \in L_\infty $.

I would like to either prove or disprove that the measure of $|g| = ||g||_\infty$, with $g \in L_\infty $, is greater than zero. I'm thinking that I should use the definition of $||g||_\infty$= ...
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0answers
36 views

Show that minimum and maximum are contained in $V(\mathcal{R})$/ Stones' axiom

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Consider $$ V:=V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, ...
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I cast doubt on the theorem about function's measurability of semi-continuous function.

Def) $f$ is measurable if, for all $a$ in $\mathbb{R^1}$, $$\left\{\mathbf{x}:f(\mathbf{x})\gt a\right\} \text{ is measurable}~.$$ Def) $f$ is said to be upper-semi continuous if ...
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1answer
41 views

Prove a.s. convergence of random variables.

I need to prove this: Assume that you have a probability space $(\Omega, \mathcal{F},P)$, $X_t$ is a stochastic process which is jointly measurable with respect to $\mathcal{B}(\mathbb{R})\times ...
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20 views

what is the relation between X and ω

From the definition of random variable: In the special case of probability space (Ω, F, P), we use the phrase random variable (RV) to mean a measurable function, that is, X : Ω → R is a random ...
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1answer
31 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
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1answer
18 views

I do not understand in a process of proving that $|H-E|=|Z|=0$ iff $|E=H-Z|$?

Notation $|E|_e$ is the outer measure of $E$. $|E|$ is the measure of $E$. A type $G_\delta$ means countable intersection of open sets. The theorem is $$ E \text{ is measurable if and only if } ...
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1answer
34 views

determining if a tail event

I am to determine if $$\{\sup X_n < \infty \}$$ is a tail event, the solutions are as follows: I don't understand how they got the line of equalities, specifically the last one, and why it holds ...
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0answers
11 views

Region under the graph of an unsigned measurable function is measurable

This was taken out of Tao's book on Measure theory: Let $f : \mathbb{R}^d \rightarrow [0, + \infty]$ be an unsigned measurable function. Show that the region $ \{ (x,t) \in \mathbb{R}^d \times ...
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18 views

Lusin theorem in measurable set $\Omega$ of $\mathbb{R}^N$

I have a question about Lusin theorem : We both know that Lusin theorem proof in X is locally compact Hausdorff space . My question is : "Can we change X into $\Omega$ ? " with $\Omega$ is ...
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1answer
14 views

Proof of a variation of the monotone convergence theorem where fn<f but fn isn't necessarily increasing.

That's basically all of it. $f_n$ and $f$ are all measurable and non-negative, $f_n\to f$ and $f_n\le f$, i want to prove that $\int_Rf=lim{\int_Rf_n}$ for $n\to \infty$ (Lebesgue integral). I know ...
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23 views

Limit of integrals is zero

Let $\lambda$ be a lebesgue integral on $[0,1)$. Define the intervals $I_{n,i}=\left(\frac{2i}{2n}, \frac{2i+1}{2n}\right)$ and $J_{n,i}=\left(\frac{2i+1}{2n}, \frac{2i+2}{2n}\right)$ for $0\leq i\leq ...
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1answer
19 views

Prove $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k 0$.

Let $\mathcal{Q_k}=[-k,k]^n\subset \mathbb{R^n}$ for all $k\in\mathbb{N}$, the n-dimensional cubes, and $f$ any integrable (lebesgue) function. Prove that $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k ...
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0answers
42 views

Continuous strictly increasing function with derivative infinity at a measure 0 set

Let $E\subset [0,1]$ with $\mu(E)=0$. Does there exist a continuous, strictly increasing function $f$ on $[0,1]$ so that $f'(x)=\infty$ for all $x\in E$ (in Lebesgue sense)? I think there exist such ...
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1answer
21 views

Convergence on $L_p$ spaces

I am trying to justify a simple result on convergence over $L_p$ spaces. The lemma is the following: Let $1\leq p<\infty$ and $0\leq f_k\nearrow f$ be measurable functions. Then $f_k\rightarrow ...
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1answer
19 views

$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$

Let $A \neq \varnothing$ and $\cal {F}$$(A) = \{F \subset A \mid F$ is finite$\}$. Define $\ell_2 (A) =L^2(A, 2^A, \mu_C)$, with $\mu_C$ the counting measure. Let $C(A) = \{f: A \to \Bbb C, \exists ...
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1answer
28 views

Decreasing sequence of non-negative Lebesgue measurable functions and MCT

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem: Suppose that $f$ and $f_n$ are nonnegative measurable ...
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1answer
19 views

What is the limit of the lebesgue integral of the function sequence fn=1/n

If $f_n=1/n$ then what is the value of the following limit (Lebesgue integral): $$\lim_{n\to\infty}\int_Rf_n$$ I basically want to prove that a generalisation of the monotone convergence theorem ...
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0answers
9 views

Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...