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

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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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26 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
13 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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27 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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41 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?
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91 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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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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14 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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69 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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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
52 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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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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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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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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57 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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28 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 ...
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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 ...
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49 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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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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32 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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1answer
21 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
29 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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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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56 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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19 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 ...
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28 views

limsup of measurable functions is not 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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24 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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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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42 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 \{ ...
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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 ...
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2answers
39 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 ...
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1answer
49 views

Some subtle questions in measure theory

This is the definition of 'almost everywhere' in Folland Real Analysis. If $(X,\mathcal{M},\mu)$ is a measure space, a set $E \in \mathcal{M}$ such that $\mu(E) = 0$ is called a null set. By ...
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$f$ is Riemann integrable iif the set of discontinuous points of $f$ has Lebesgue measure zero

This is a well known result in mathematics, but it's my first time attempting to prove it. I'm following the second book of Analysis from Folland. Below are the notations used and the theorem, from ...
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20 views

Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
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I've followed the proof in the book - it seems more difficult than it has to be (basic measure theory, more real analysis)

This basically involves summations, so I'm tagging it real analysis. If $A\subset B$ then $\mu(A)\le\mu(B)$ - take this as fact, and $\mu:S\rightarrow\mathbb{R}\cup\{+\infty\}$. So a function that ...
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Show that $S$ is a $\sigma$-algebra

Let $C$ be any class of subsets of a set $X$. Let $Y \subset X$. Let $A(C)$ be the algebra generated by $C$. Let $S:=\{E \cup(B \cap Y^{c})\mid E \in S(C \cap Y)$, $B \in S(C)$}. Show that $S$ is a ...
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borel-measurable function is pointwise limit of a sequence of continuous functions, wich is uniformly bounded

Let $H$ be a Hilbert space over $\mathbb{C}$, $A\in L(H)$ ( $A:H\to H$ is linear and continuous) and let $A$ be self-adjoint. Consider the spectrum of A, $\sigma(A)$ and $f:K\to \mathbb{K}$ a bounded, ...
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Show that $S(f^{-1}(C))=f^{-1}(S(C))$

Show that $S(f^{-1}(C))=f^{-1}(S(C))$, where $f:X \to Y$ is a function and $C$ is a non empty family of subsets of $Y$. And $f^{-1}(C):=\{f^{-1}(c)|c \in C\}$ and $S(C)$ is the $\sigma$-algebra ...
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10 views

Is the weak limit of a sequence of translation-invariant measures itself translation-invariant?

Let $(X,\mathcal{B})$ be a measure space and $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of translation-invariant measures on it that converges weakly to the measure $\nu$ on $(X,\mathcal{B})$. ...
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29 views

Is a function from cantor set to R is measurable?

Is a function f from E to R is measurable? when E is cantor set. We are only known that the measure of E is 0.
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Exercise of Measurable Functions [on hold]

Let ($\mathbb{N}$, $\mathcal{P}(\mathbb{N})$) and $f(n)$= $\tfrac{(-1)^{n+1}}{n} $ with $n\in \mathbb{N}$, I need to prove that this function is measurable or not Could give me some suggestions
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1answer
17 views

Sequence of continuous functions convergent to $0$ with the integral equal to $1$

I am looking for a sequence of continuous functions $\{f_m\}$ defined in $A\subset\mathbb{R}$ with $\lim\limits_{m\to\infty} f_m=0$ such that $\int_A f_m \;d\mu=1$. The problem I have is with the ...
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Convolution of two indicator functions can't be constant

Let $A,B \subset S^1$ be measurable sets (considering $S^1$ with say the lebesgue measure). I'm trying to prove that if the convolution $1_A*1_B$ is constant then one of $A$ or $B$ is a full measure ...
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22 views

Fatou's Lemma and Counting Measure

I have a vague problem in a Measure and Integration book here. They ask me to consider $\mu$ the counting measure in $\mathbb{N}$ and interpret Fatou's lemma, monotone and dominated convergence ...
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1answer
47 views

Measure on compact

I am tryying to solve exercise: Let $K \subset R$ is compact with no isolated point. Find Borel nonatomic measure $\xi$ on $R$ such that $\operatorname{supp}(\xi)=K$ Thank's on all help.
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1answer
49 views

Convergence of $f_n(x) = 2^n \cdot F(2^n (x-a_n))$ with $F(x) = e^{-x^2}$ with different notions of convergence.

I had my measure theory exam this morning, and one exercise was the following: I really can't see a solution. During the semester, we talked about almost everywhere convergence, almost uniform ...
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20 views

Restating a $\ge$ constraint as a limit involving $>$

Restating a $\ge$ constraint as a limit involving $>$ I've countered this before, long ago, but I've tried and cannot remember what I did. I am trying to solve something of the form that goes: ...
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1answer
15 views

What can we say about the absolute continuity of the joint distribution of $(X,Y)$?

If the distribution of $X$ is absolutely continuous with respect to the counting measure, and distribution of $Y$ is absolutely continuous with respect to the Lebesgue measure, what can we say about ...