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

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Generating Borel algebra - proof

In the second paragraph of the proof it says: "To prove that $\mathcal{B}(\mathbb{R})$ is also generated by the other classes of intervals, it suffices to prove that any interval $]a,b[$ is contained ...
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0answers
10 views

reducible measure functions

Let $f:\Bbb R^n \rightarrow \Bbb R^n$ be a nonexpansive map, that is $\|f(x) - f(y)\| \leq \|x-y\|.$ I want to know that if $f$ reduces measure, i.e. if $A,f(A) \subseteq \Bbb R^n $ are measurable, ...
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12 views

Measure Theory Book for My Background / Need

My current Math background is as follows: 1) Read first 7 chapters of Rudin "Principles of Mathematical Analysis" and solved a lot of the given problems. 2) Completed Munkres "Analysis on Manifolds" ...
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5 views

Measure in a locally compact Hausdorff space

Suppose that $X$ is a locally compact Hausdorff space in which every open set is $\sigma$-compact and such that for every compact subset $K$ of $X$ we have $\mu(K) < \infty$. What can we deduce ...
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1answer
27 views

$\sigma$-algebra generated by all supersets of $[0, 1]$ [on hold]

I am trying to find the $\sigma$-algebra generated by all subsets $B$ of $\mathbb{R}$ that contain the set $[0, 1]$, i.e., $[0, 1] \subseteq B \subseteq \mathbb{R}$. Any help would be appreciated.
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1answer
113 views

Is any subset of the Cantor set a Borel set?

As far as I know, the Cantor set is a Borel set because it is the union of a countable collection of closed sets. Now is any subset of the Cantor set a Borel set?
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0answers
19 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
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0answers
13 views

Clarification on product space, conditional probability measures

Suppose we have two random variables $X_i$ on the probability space $(\Omega_i, \sigma(\Omega_i),P_i)\; i = 1,2$. Now, $P_i$ is just a technical tool and we consider directly the distribution of $X_i$ ...
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0answers
17 views

If the support of a function is contained in a Borel set, is the support of its derivative also contained there?

Let $f$ be a function such as $\operatorname{supp}(f)\subset Q$ where $Q$ belongs to the Borel $\sigma$-algebra on $\mathbb{R}^d$ Do we have $\operatorname{supp}(f')\subset Q$?
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3answers
35 views

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$. I just started learning about measure this week, so I don't know any theory about measure except the definition of outer measure ...
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13 views

If $\mu$ is a semifinite measure, every set of infinite measure contains a subset of arbitrarily large, finite measure [duplicate]

I need help in this question. Thank you. If $\mu$ is a semifinite measure and $\mu(E)=\infty$, for $C>0$ there exists $F\subset E$ with $C<\mu(F)<\infty$.
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1answer
17 views

$\lim_{n \to \infty} \int_{\Omega}X_n d \mu = +\infty$ under some conditions

Suppose $X_n$ are measurable functions in $L^1$ defined on the measure space $(\Omega, \mathfrak{F}, \mu)$. Suppose that $0 \leq X_n$ a.e. for all $n$ and $X_n \leq X_{n+1}$ a.e. for all $n$. Thus ...
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0answers
12 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
2
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1answer
28 views

right continuous representant of L^1 function

Would someone know if a function in $L^1(\Omega)$ or more generally in $L^p(\Omega)$ must have a right continuous representant. $\Omega$ being a open set of $\mathbb{R}$. Thankfully,
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2answers
47 views

Show that $(\mathcal{M},d)$ is complete metric space

Let $(\Bbb{R},\mathcal{M},\mu)$ be the Lebesgue measure space modulo the equivalence relation $A\sim B$ if $\mu(A\bigtriangleup B)=0$. Let $d(A,B)=\mu(A\bigtriangleup B)$. Show that $(\mathcal{M},d)$ ...
2
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1answer
38 views

Struggling with something 4 sources use “clearly” with (involving $\inf$ of a set)

This is technically measure theory but it as much real analysis. We have a Measure $\mu:R\rightarrow\mathbb{R}_{\ge0}\cup\{\infty\}$ We define the outer measure $\mu^*$ to be: ...
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0answers
21 views

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 $E \in \mathbb{R}$. ...
1
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1answer
33 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 ...
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0answers
29 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
13 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 ...
3
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1answer
35 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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15 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
16 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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0answers
21 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
19 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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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
40 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 ...
1
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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
21 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
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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0answers
39 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
86 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
68 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
47 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
20 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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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
25 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
51 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
22 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 ...