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

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25 views

looking for help with convergence in measure problem

Why is it true that if $(X, \mu)$ is a finite space, $f_n \to f$ in measure, and for each $n$ and $\epsilon > 0$ there exists $\delta > 0$ such that $\mu(E) < \delta \Longrightarrow \int_E ...
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0answers
36 views

Union of uncountable family of $\sigma$-algebras

Suppose that $\{F_\alpha\}$ is an uncountable family of $\sigma$-algebras, and let $H=\bigcap_\alpha F_\alpha$. Is $H$ also a $\sigma$-algebra? Why or why not? I understand that the intersection ...
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0answers
15 views

Ordering on probability measures implies equality

Let $\Omega$ be a non-empty set, $\mathcal{F}$ a $\sigma$-algebra of subsets of $\Omega$ and $P,~Q$ two probability measures on $(\Omega, \mathcal{F})$. Assume that $P(A)\leq Q(A)$ for all $A\in ...
2
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0answers
62 views

Will the generated sigma algebra have this property?

Lets say you have a measurable space $(\Omega, \mathcal{A})$. And a measurable function $X: (\Omega, \mathcal{A})\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$. We then know that for the sigma ...
3
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2answers
46 views

For a.e. $x \in [0, 1]$, there are finitely many $p/q$ such that $\left| x - p/q \right| < 1 / \left( q \log q \right)^2$

I am stuck on a qualifying exam problem and was hoping to get some help. Show for a.e. $x \in [0, 1]$ that there are finitely many $p/q \in \mathbf{Q}$ in reduced form such that $q \geq 2$ and ...
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0answers
19 views

When $X_{n\wedge N}$ converges to $X_N$ in probability for martingale $X_n$ and stopping time $N$?

Suppose $\sigma$-algebras $\{\mathcal{F}_n\}$ is a filtration and random variables $\{X_n\}$ are adapted to $\{\mathcal{F}_n\}$. $N$ is a stopping time w.r.t $\{\mathcal{F}_n\}$. If $(X_n, ...
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1answer
34 views

Exercise about measurable and continuous functions

I want to propose to you this exercise. Let $f:[0,1]\times \mathbb{R}\to\mathbb{R}$ a function with these properties: 1)For every $x\in\mathbb{R}$ the map $t\mapsto f(t,x)$ is measurable. 2)For ...
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0answers
22 views

Convergence in measure : some questions.

I know that $f_n\to f$ in measure if $$\forall \varepsilon>0, \lim_{n\to\infty }m\{x\mid |f_n(x)-f(x)|>\varepsilon\}=0.$$ Does it mean that: Q1) $f_n\to f$ in measure if $$m\left\{x\mid ...
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1answer
35 views

If $f_n\to f$ in measure, is there a subsequence s.t. $f_{n_k}\to f$ a.e.? [duplicate]

If $f_n\to f$ in measure, is there a subsequence s.t. $f_{n_k}\to f$ a.e. ? The convergence in measure is a little be abstract to me, I don't really see what it means (even if I know the definition). ...
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0answers
16 views

Convergence of a sequence of measure distribution functions

I'm trying to show that for $f$ and $f_n$ measurable, $|f_n|\uparrow|f|$ implies $d_{f_n}\uparrow d_f$, where $d_f(\alpha)$ is the distribution function given by $$d_f(\alpha):=\mu(\{x: ...
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1answer
27 views

Integration respect to two measures

Let $(X,\mathcal{K},\mu_1)$ and $(X,\mathcal{K},\mu_2)$ be two measure spaces, and let $\mu=\mu_1+\mu_2$. Assume $f:X\to[0,+\infty]$ are integrable both respect to $\mu_1$ and $\mu_2$, how do we ...
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0answers
25 views

Is every strongly measurable function to $\mathbb{R}$ also measurable?

Here seems a simple enough proof of the statement: Let $f: X \to \mathbb R_{≥0}$ be measurable. Construct $$f_n(x)=\sum_{k=0}^{n^2}\frac{k}{n}\cdot ...
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0answers
26 views

What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
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3answers
15 views

Outer Measure definition

In the definition of Lebesgue outer measure/ outer measure , m*(A) = inf [Σ l(In)] Here how can one take infimum over a summation? Please elaborate.
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1answer
28 views

Random function of random variable

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Suppose that $\phi:(\Omega,\mathbb{R})\rightarrow\mathbb{R}$ is such that $\phi(\omega,\cdot)$ and $\phi(\cdot,x)$ are Borel measurable. ...
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1answer
25 views

Almost everywhere equality of r.v.'s , based on information on mean values.

Let $X, Y$ be two random variables on a probability space $(\Omega, \mathcal{F},P)$, where $\mathcal{F}=\sigma(\mathcal{E})$. We assume that: $\mathcal{E}$ is closed on intersections, i.e. $A\cap ...
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1answer
31 views

Does this sum of normally distributed random variables necessarily result in a continuous R.V?

Originally I had asked whether two continuous random variables can sum to a discrete random variable. More specifically, I am wonder whether, if we Let $X_n \sim \text{iid } N(0,\sigma_x^2)$ and $Y_n ...
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0answers
30 views

Real Analysis, Folland problem 3.2.14 The Lebesgue - Radon-Nikodym Theorem

Relevant background information: We say that two signed measures $\mu$ and $\nu$ on $(X,M)$ are mutually singular if there exists $E,F\in M$ such that $E\cap F = \emptyset$, $E\cup F = X$, $E$ is ...
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0answers
46 views

Understanding the concept of measurability of random variables

If a random variable $X$ is $\mathcal{F}_{t_0}$-measurable, where $\{ \mathcal{F} \} _{ t \geq 0}$ is an underlying filtration, does that mean that from the time $t_0$ onwards, the random variable $X$ ...
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0answers
35 views

Prove that this function is in $L^\infty$ with $\lVert g\rVert_\infty \le C$.

My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise. Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in ...
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1answer
42 views

Continuous Function on a Set With Content Zero

I am trying to prove a proposition about a continuous function over part of a compact set, and I have gotten stuck. The proof will be completed if I can verify the following: If $f$ is a continuously ...
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1answer
55 views

Never seen this notation before: $\int (y-f(x))^2 Pr(dx,dy) $

I have never seen an integral like this: $$\int (y-f(x))^2 Pr(dx,dy) $$ What is that? More precisely what is $Pr(dx,dy)$? And how is that integral defined? I found it in Elements of Statistical ...
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0answers
17 views

“Equidecomposable”: informal meaning

I am having trouble understanding the definition of the term "equidecomposable". Is it like two sets are split into many sets and then these many sets can be joined together to make either of the two ...
2
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2answers
41 views

Lebesgue Integral of a non negative piecewise function

Consider the function over [0,1] given by $f(x)= \begin{cases} 0 & x \in \mathbb{Q}\\ x & x \notin \mathbb{Q} \end{cases}$ In order to compute the Lebesgue integral of $f$ we need to find an ...
2
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0answers
56 views

How to understand $E(X\mid B)$ in the measure theory way

From undergraduate probability course, we learn $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ given $P(B)>0$. And we learn that if $(X,Y)$ has a joint density $f(x,y)$, we can calculate marginal density ...
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1answer
15 views

Real Analysis, Folland Problem 3.3.20 Complex Measures

Related definitions - A complex measure on a measurable space $(X,M)$ is a map $\nu: M\rightarrow\mathbb{C}$ such that i.) $\nu(\emptyset) = 0;$ ii.) if $\{E_j\}$ is a sequence of disjoint sets in ...
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1answer
41 views

Book Recommendation for Measure Theory in n-Space

What's a standard book on multidimensional measure theory? I'm aware of some books on functions of several variables, but they do not discuss measure theory or Lebesgue integration in space. Thanks. ...
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1answer
58 views

Integration over finite partition of integration domain

I think the title does not reflect my problem very well. Feel free to leave a comment with a more appropriate title. Let $f \in L^1([0,1])$. How do I prove there exists a partition of $[0,1]$ into ...
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0answers
21 views

If a function is continuous and finite almost everywhere, does it follow that it is finite everywhere? [closed]

If a function is continuous and finite almost everywhere on its domain with respect to Lebesgue measure, does it follow that it is finite everywhere?
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2answers
35 views

Why a cover of a set exist?

In the definition of the measure, we have that $$m^*(E)=\inf\left\{\sum_{i=1}^\infty |Q_i|\mid E\subset\bigcup_{i=1}^\infty Q_i\right\}$$ where $Q_j$ are closed cube. My question is : Why for any $E$ ...
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0answers
28 views

Getting the independent variables from dependent variables. [duplicate]

This question is related to the solution in the answer here: Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. Quick description of my problem: Let ...
2
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1answer
32 views

Prove that $\int g(x)dx=\int f(x)dx$.

Let $f:[0,b]\longrightarrow \mathbb R$ and $g:]0,b]\longrightarrow \mathbb R$ define as $$g(x)=\int_x^b\frac{f(t)}{t}dt.$$ Prove that $g$ is integrable and that $$\int g(x)dx=\int f(x)dx.$$ So ...
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1answer
21 views

Integration of a measurable function

Let $\phi(x)$ be a simple function. If $a_1, a_2, . . . . , a_n$ are the distinct values taken by $\phi$ and $A_i = [x : \phi(x) = a_i]$, then $\phi(x) =\sum_{i=1}^n a_i \Large {\chi}_{A_i}$ $(x)$ , ...
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37 views

Relation of absolute convergence to expected value.

Claim: If $X_n \overset{\text{a.s.}}{\longrightarrow} X$ then $\mathbf{E}[\lim_{n\to\infty}X_n] = \mathbf{E}[X]$. Question: Is this true? Below is a proof, but I'm worried that I made a mistake. ...
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0answers
28 views

Is the following modification of a martingale still martingale?

I have a following question. Let $Z$ be a Geometric Brownian motion, $\frac{dZ(t)}{Z(t)} = \omega dt + \sigma dW(t) $ For $\omega = -\frac{1}{2}\sigma^{2}$ one can proof that $Z$ is a martingale. ...
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0answers
46 views

Convolution of measures, why is the notation like this?

In both my book, and on Wikipedia they define convulution of two measures like this: $(\mu_1*\mu_2)(B)=\int_{\mathbb{R}^d}\mathcal{X}_B(x+y)d\mu_1(x)d\mu_2(y)$ It doesn't seem like a typo, but ...
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1answer
27 views

$\mu \ll m$ finite Borel implies $x \mapsto \mu(A + x)$ is continuous

Why is it true that if $\mu$ is a finite Borel measure on $\mathbf{R}$ which is absolutely continuous with respect to Lebesgue measure $m$, then $x \mapsto \mu(A + x)$ is continuous for any fixed ...
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0answers
31 views

Real Analysis, Folland problem 3.3.18 Complex measures

Related definitions - A complex measure on a measurable space $(X,M)$ is a map $\nu: M\rightarrow\mathbb{C}$ such that i.) $\nu(\emptyset) = 0;$ ii.) if $\{E_j\}$ is a sequence of disjoint sets in ...
1
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0answers
67 views

Why is linearity a requirement of a integral

I was reading Philip Protter's Stochastic Integration and Differential Equations textbook. He mentions that an operator, $I_X$, induced by $X$ should be linear to be called an integral. I have a ...
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1answer
35 views

Show that $l^p \subseteq l^q$ for $1 \leq p < q < \infty$

$$l^p = \{ (a_k)_{k \geq 1} : \sum \limits_{k=1}^{\infty} |a_k|^p < \infty \}$$ Since it is said $l^p \subseteq l^q$, I would have thought we have to show $$\sum \limits_{k=1}^{\infty} |a_k|^q ...
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0answers
20 views

Showing that $I(\xi_1,…,\xi_d)=0$

Let $\xi_1,...,\xi_d \in S^{d-1}$ and $Leb(B)=0$. We define $$I(\xi_1,...,\xi_d) = \int_0^\infty \cdots \int_0^\infty 1_B (r_1 \xi_1+\cdots+r_d \xi_d) \prod_{j=1}^d g(\xi_j,r_j) (r_j^2 \wedge 1) ...
3
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2answers
169 views

Applications of Dominated/Monotone convergence theorem

Consider a measure $\mu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Consider the function $f: [0,\infty)\rightarrow ...
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2answers
30 views

Proving something is a norm

Let $a<b$ be real numbers and $X=C[a,b]$ be the space of continuous functions $f:[a,b] \rightarrow \mathbb R$ Prove that $||f||_1 = \sup _{t \in [a,b]} |f(t)|$ indeed defines a norm on $X$. The ...
3
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1answer
51 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
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2answers
13 views

Regarding sigma fields and its subsets [closed]

There is a subset of sigma field $G_2$, say $G_1 \subset G_2$. $G_1$ is proven to be a sigma field. Does this necessarily imply that $G_1 = G_2$?
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1answer
18 views

Subsequence of $L^{2}(\Omega)$ - bounded sequence weakly * converging to a measure

I was reading a well available article in the internet: "THE COMPENSATED COMPACTNESS METHOD APPLIED TO SYSTEMS OF CONSERVATION LAWS by Tartar"; there at Page-266, it is written: "$L^{1}(\Omega)$ is ...
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1answer
34 views

Can one divide a set into subintervals [duplicate]

Sorry that the question title is unclear, I didn't know how to ask it. Take set $A \subseteq [0,1]$, measurable. Does there exist a sequence $x_1,x_2,\dots$ such that $\forall x_i$, \begin{align*} ...
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0answers
19 views

Deriving densities with respect to another density [closed]

Consider three probability measures $P,Q,\mu$ defined on the same measurable space $(\Omega,\mathcal{F})$. Suppose $P,Q$ have densities $p,q$ with respect to $\mu$, i.e $p=\frac{dP}{d\mu}$ and ...
1
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0answers
42 views

Integral Inequality (CDFs and PDFs)

Suppose I have a function $g \geq 0$ defined by $$g(x) = \int_{-\infty}^{x}f(t)\text{ d}t \geq 0\text{, }x \in \mathbb{R}\text{. }$$ I know for a fact that $g$ is continuous and nondecreasing. Is ...
0
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2answers
44 views

Is there a shorter proof for this variant of the Dominated Convergence Theorem?

I finally managed to proof this variant of the Dominated Convergence Theorem: Theorem (Variant of Dominated Convergence Theorem). Let $f, f_k: X \to \overline{\mathbb R}$ be $\mu$-measurable, $g, ...