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

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measure and increasing functions

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ incresing, bounded, continious, and $\mu$ corresponding measure($\mu(-\infty,x)=f(x))$. Prove that $ \mu(E)=m(f(E))$ for all Borel set $E \in \mathbb{R}$. ...
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1answer
15 views

Convergence in measure - product

I'm trying to prove the following statements in Folland's book. Let $(X,\mathcal{M},\mu)$ be a measure space. If $f_n\to f$ in measure and $g_n\to g$ in measure, then $f_n+g_n\to f+g$ in measure and ...
4
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0answers
18 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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0answers
7 views

image or projection of a universally measurable set

Suppose $A \subset [0,1]\times[0,1]$ is universally measurable. Is it true that its projection to the first coordinate is a universally measurable subset of $[0,1]$? What is known is that the ...
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0answers
24 views

Prove that measure of $A$ is $1$

Let $A\subset (0,1)$ be a Lebesgue measurable set and $\lambda>0$. Suppose that if $0\le a<b\le 1$ then $\mu(A\cap (a,b))\ge \lambda(b-a)$. Prove that $\mu(A)=1$. It is clear that $\lambda \le ...
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0answers
13 views

Lebesgue Measure of a set satisfying infinitely many solutions of this inequality

I am trying to find the following. Suppose that $\alpha_k > 0$, and $\sum \alpha_k < 0$. Let's consider the set $$A = \{x\in(0,1) | \hbox{the inequality} |x -{p \over q}| < {\alpha_q \over ...
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1answer
15 views

Complex Borel Measure Decomposes as Discrete + Continuous Measures?

I am trying to prove that any complex Borel measure $\mu$ on a measure space $(X, \mathfrak{M})$ decomposes into the sum of a discrete measure $\lambda$ and a continuous measure $\nu$, $\mu = \lambda ...
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17 views

Existence of regular conditional distribution of random variable given the value of another variable

Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space with a measurable function $Y: (\Omega, \mathcal{A}) \rightarrow (E, \mathcal{E})$ and another measurable function $X: (\Omega, ...
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1answer
12 views

Is the product measure space generated by the filtration adapted to the projection maps?

Let $(\Omega, \mathcal A)$ be a measure space. Consider the product measure space $(\Omega^{\mathbb N}, \mathcal A^{\mathbb N})$ and denote by $\pi_n : \Omega^{\mathbb N} \to \Omega$ the $n$-th ...
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3answers
35 views

Let $S = \{(a,b] \cup [−b,−a) : a < b\text{ are real numbers}\}$. Show that $\sigma(S)$ is smaller than the Borel $\sigma$-algebra of $\mathbb R$.

Let $S = \{(a,b] \cup [−b,−a) : a < b\text{ are real numbers}\}$. Show that $\sigma(S)$ is smaller than the Borel $\sigma$-algebra of $\mathbb R$. How can I find a Borel set which is not in ...
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0answers
16 views

Generic rank of tensors

Let the tensor product of the type $$ \underset{k=1} { \overset{m} \bigotimes } v_k$$ denote a simple tensor. As underlying fields, take $$ \underset{k=1} { \overset{m} \bigotimes } ...
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0answers
24 views

Borel Sets and relation to probability theory.

I am currently having difficulty understanding the link between Borel Sets and Probability theory. How/Why are Borel Sets used in Probability theory?
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2answers
36 views

Question about Rudin's example of topological space

I began reading Rudin's Real and Complex Analysis, and I have a question about the following: Rudin defines a topology $T$ in a set $X$ as the collection of subsets of $X$ such that (i) empty set ...
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1answer
35 views

Measurability of functions

I have to check that $g:(0,1)\to \mathbb{R}$ is measurable. I known there exists a $C^2$ convex function $F:(0,1)\to \mathbb{R}$ (with $F''\geq 1$), and measurable functions $v,u:(0,1)\to (0,1)$ ...
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0answers
18 views

Density of product of random variable

I would like to derive the product density of two independent continuous random variable in a measure theoretic framework. I am well aware of the result which can be found here: ...
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0answers
10 views

For any $A \subseteq \mathbb{R}^d$, there exists a $G_\delta$ set $H \supseteq A$ such that for every measurable E, $|A \cap E|_e = |H \cap E|$

For any $A \subseteq \mathbb{R}^d$, there exists a $G_\delta$ set $H \supseteq A$ such that for every measurable E, $|A \cap E|_e = |H \cap E|$ I've done the case that $|A|_e < \infty$ using ...
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1answer
25 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
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0answers
38 views

Origin of $\sigma$-algebra

In what paper, article or book was the notion of an $\sigma$-algebra first defined or mentioned? Or at least how far could this concept traced back?
3
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2answers
83 views

Finding integral over inconvenient set

Put $F = \{ (x,y) \in \mathbb{R}^2 : |x^2-y^2| \leq 1, 2|xy| \leq 1 \}$. How do we find the following integral? $$\int_F (x^2 + y^2) \,d(x,y)$$ I'm sure we need to use Jacobi's transformation ...
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1answer
24 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
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0answers
13 views

Sigma field generated by the union of a field and a set

I am trying to show that; If $H$ is a set lying outside a field (or $\sigma$-field) $\mathcal{F}$, then the field (or $\sigma$-field) generated by $\{\mathcal{F}\cup\{H\}\}$ consists of sets of the ...
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1answer
13 views

limit, measure, and 1 sided continuity

Let $(R,B(R),m)$ be a finite measure. For function $F, G$ on $\mathbb{R}$, define $F(x) = m((-\infty,x])$ and $G(x) = m((-\infty,x)) $ 1) Prove that $F$ is right continuous while $G$ is left ...
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2answers
65 views

Can I use lebesque dominated convergence?

Calculate the following: $$\lim_{n \rightarrow \infty} \int_{0}^{\infty} \exp(−nx(\sin(x))^2)\,dx$$ my idea was using $f_n(x) = \exp(−nx(\sin(x))^2) < 1$ but $1$ is not integrable under ...
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1answer
16 views

Prove that the measure of the set is equal to the measure of intersection of its members

Given any set $ A \subseteq \mathbb R $, there is a set $G$ that contains $A$ such that $$ G = \cap _ {i=1} ^ \infty A_i $$ with each $A_i$ open and $m*(G) = M*(A). Here is my approach so far: ...
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1answer
20 views

Show countable additivity of a certain probability measure

Let $\mathcal{F}$ be the field consisting of the finite and the co-finite sets in an infinite and ${\bf{uncountable}}\;\Omega$, and define a probability measure $P$ on $\mathcal{F}$ by taking $P(A)$ ...
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2answers
44 views

If $X$ is a random variable with distribution $\mu$, prove $\int \limits_{\Omega} X(\omega) \, dP(\omega) = \int \limits_{\Bbb R} x \, \mu(dx)$.

I'm trying to prove $\int \limits_{\Omega} X(\omega) \, dP(\omega) = \int \limits_{\Bbb R} x \,\mu(dx)$ if $X$ is a random variable defined on $(\Omega, \mathcal{F}, P)$. I understand how to prove ...
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2answers
19 views

When does probability mass outside a sufficiently large ball is small?

Many times when I read books about statistics or probability theory, I encounter proofs which said: For any $\epsilon>0$ there is an $M\in(0,\infty)$ such that $\text{Pr}\{X\in ...
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0answers
24 views

question 3.29 from Folland Real Anyalysis

If $F$ in $NBV$ is real-valued, then show $u_F ^+=u_P$ and $u_F ^-=u_N$ where $P$ and $N$ are the positive and negative variations of F. (Use Exercise 3.28) Source: Folland Real analysis exercise ...
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1answer
33 views

question 3.40 from Folland Real Anyalysis

Let $F$ denote the Cantor function on $[0, 1]$ (see $§1.5$), and set $F(x)= 0$ for $x<0$ and $F(x)=1$ for $x>1$. Let ${[a_n, b_n]}$ be an enumeration of the closed subintervals of $[0,1]$ with ...
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0answers
29 views

Prove that S=$\cup_{\varepsilon \ge0} \cap_{n=1}^{\infty}\cup_{m=n}^{\infty}E_m(\varepsilon)$ [on hold]

Let {$f_n$}$_n$ and $f$ be real valued function defined on $\mathbb R $ for $\varepsilon$>0 and $m\in \mathbb N$, define $E_m(\varepsilon)=${$x\in \mathbb R : |f_m(x)-f(x)|\ge \varepsilon$}. And let ...
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0answers
23 views

proving that the graph of a function is of Jordan measure zero

Let $f$ be an integrable function from $B$ to $[0,\inf]$ where $B$ is a sphere in $\mathbb{R^n}$. Exercise: For $f$ and $B$, the graph $$ \Gamma=\{(x,f(x)):x\in B\} \subset \mathbb{R}^{n+1} $$ is of ...
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2answers
50 views

Show that for any $1\leq p<\infty$, the set $L^1\cap L^p$ is a dense subset of $L^p$

Show that for any $1\leq p<\infty$, the set $L^1\cap L^p$ is a dense subset of $L^p$. Let $f\in L^p-L^1$. We need to find a sequence $\{\phi_n\}_n$ in $L^1\cap L^p$ converging to $f$. And I ...
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2answers
21 views

Steiner symmetrization of Lebesgue measurable set

I'm reading a proof in "Evans / Gariepy: Measure theory and fine properties of functions" of the following statemant: Theorem: Let $A \subset \mathbb R^n$ be $\lambda^n$-measurable, $a \in \mathbb ...
1
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1answer
9 views

Meaning of sampling i.i.d rvs from (random) probability measure?

Quote from book: "Consider an arbitrary atomic probability measure $\Gamma$ on unit sphere. Let $(\sigma_{l})$ denote an i.i.d sample from $\Gamma$." I don't understand the second sentence. Does it ...
3
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1answer
46 views

How many points are needed to intersect all elements in a sequence of measurable sets

Suppose $(X,\mathcal B, \mu)$ is a probability space and $n\in\mathbb N$ is an arbitrary but fixed integer. Is it true that if $m\in\mathbb N$ and if $A_1,\ldots,A_m\in\mathcal B$ with ...
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1answer
24 views

Need help with convergence of functional sequences [on hold]

a) Does the sequence of functions $f_n=-{1\over n} χ_{[0,n]}$ converges to $0$? Which condition from Fatous theorem isn't satisfied? b) Can we apply Lebesgue's Dominated Convergence Theorem on this ...
1
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1answer
25 views

What can you say about union of two non measurable set. They are measurable or not?

What can you say about union of two non measurable set. They are measurable or not? Is it necessarily true?Thinking about α-cantor set I wonder if the complement of a non measurable set is a non ...
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0answers
30 views

Discrete measures and discrete kernels

This question was also posted here. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...
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How to solve this? Please Help me. [on hold]

How to show that the outer measure of singletone set is zero?
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1answer
17 views

Question in the proof of Caratheodory's theorem

First, let me recall Caratheodory's theorem: Theorem (Caratheodory). Let $\mathcal{M}:=\{E:E\subseteq X,\theta A=\theta(A\cap E)+\theta(A\cap E^c),\forall A\subseteq X\}$ be the class of measurable ...
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1answer
28 views

Kolmogorov 0-1 law

Initial question: $X_n$, $n \in\mathbb N$, are independent real-valued random variables. Let $S_n$ be defined, for each $n\in\mathbb N$, by the sum: $S_n = X_1+X_2+...+X_n$. Prove that either the ...
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0answers
17 views

A two-stage experiment where the first stage has two independent outcomes

If $P(Y_1\in \cdot|X_1, X_2) = P(Y_1\in \cdot|X_1)$ and if $P(Y_2\in \cdot|X_1, X_2) = P(Y_2\in \cdot|X_2)$ and if $X_1$ and $X_2$ are independent, are $Y_1$ and $Y_2$ independent given $X_1, X_2$, ...
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2answers
52 views

Use Dominated convergence theorem to show that $f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3}$ is differentiable

Let $$f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3},$$ how can we show that f is differentiable everywhere by using the Lebesgue dominated convergence theorem? I know this theorem as saying ...
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1answer
16 views

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert ...
2
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1answer
17 views

limsup of measurable functions is measurable

Suppose $f_r(x)$ is measurable for any fixed $r>0$, I was wondering whether $\limsup_{r\to 0^+}f_r(x)$ is measurable. I know the limsup of sequence of measurable functions is measurable, and I ...
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0answers
20 views

Set of all probability measures with finite support

Let $X$ be an uncountable set endowed with the discrete topology. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, and consider the subset $A$ of $\mathcal{P}(X)$ consisting ...
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2answers
68 views

Determining a charge through subsets

A charge is a finitely additive set function $c: \mathcal{P}(\mathbb{N}) \to [0, 1]$ such that $c(\mathbb{N}) = 1$ and $c(\{n\}) = 0$ for every $n \in N$. Here $\mathcal{P}(\mathbb{N})$ is the set of ...
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1answer
36 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
0
votes
0answers
20 views

Measure of the Brownian Motion

Let $B_t$ be a Brownian motion and $B(x,r)$ the ball of x with radius r. Define $\mu_t(A):=\int_0^t 1_A(B_s)ds$ for $A\in \mathcal{B}(\mathbb{R})$. Now I'm not sure if the following equations are ...
2
votes
2answers
36 views

show $\int f_kd\mu\leq C$ for $f_k\geq0$, $\int fd\mu\leq C$

Let $(\Omega, \mathcal A,\mu)$ be a measure space and $f_k\rightarrow f$ a.e., $f_k\geq0$ and $\int f d\mu\leq C$ for some $C>0$. How can you show $\int f_k d\mu\leq C$ ? My attempt: I thought ...