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Does a section of a finite-measure set in the product measure space have a finite measure almost everywhere?

Let $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$ be $\sigma$-finite measure spaces, and $\mu\otimes\nu$ the product measure on the product $\sigma$-algebra $\mathcal{S}\otimes\mathcal{T}$. Let $M\...
ashpool's user avatar
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Two dimensional Borel Measure proof

A measure on $\mathcal{B}_2$ is called a two dimensional Borel measure. Suppose that $\mu$ and $\nu$ are finite two dimensional Borel measures such that $\mu(A\times B) = \nu(A \times B)$ for all $A,B\...
Joey's user avatar
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Product measure on R^2

$\int_{]a_1, b_1] \times ]a_2, b_2]} f \, d\lambda^2 = \int_{a_2}^{b_2} \int_{a_1}^{b_1} f(x_1, x_2) \, dx_1 \, dx_2$ f is continous function from R^2 to R. I have to show the equation above. My ...
vwhg1050's user avatar
2 votes
2 answers
126 views

Finite radius ball Lebesgue integral on product of Lebesgue measure

Given $n \in \Bbb{N}$, let $m^n$ denote the $n$ dimensional Lebesgue measure on $\Bbb{R}^n$. Fix $p>0$ and show if $B_r(0) \subset \Bbb{R}^n$, then if $f(x)=\vert \vert x \vert \vert^{-p}$, where $\...
homosapien's user avatar
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2 votes
0 answers
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When a set function on a product sigma-algebra is a measure

Let $(X,\mathcal{A})$ and $(X,\mathcal{B})$ be measurable spaces. Every measure $\mu$ on $(X\times Y, \mathcal{A}\otimes\mathcal{B})$ gives an assignment $\mathcal{A}\times\mathcal{B}\to\Bbb{R}$ via $(...
geodude's user avatar
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3 votes
1 answer
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Proof of the measurability of functions in Theorem 1.10 (Product measure) in "Analysis" by Lieb and Loss

I am confronted with the following problem: Let $\left(\Omega_1,\Sigma_1,\mu_1\right)$ and $\left(\Omega_2,\Sigma_2,\mu_2\right)$ be two sigma-finite measure spaces. Let $A$ be a measurable set in $\...
mathishard's user avatar
1 vote
1 answer
40 views

Probability w.r.t. sample

In FAST LEARNING RATES FOR PLUG-IN CLASSIFIERS by Audibert, Tsybakov, they use the following notation in the context of binary classification: Let $(X, Y)$ be a random couple taking values in $Z = R^d\...
newbie's user avatar
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1 vote
1 answer
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notation of sample in classification context

I have a question about the following notation: $(X,Y)\in \mathbb R^d \times \{0,1\}$ with joint distribution $P$. $X$ vector of features, $Y$ corresponding label. Now consider the sample $(X_i,Y_i)_{...
newbie's user avatar
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When is the product of Dirac delta distributions defined?

Wikipedia defines a $n$-dimensional delta function as just taking $$\delta (\mathbf{x}) = \prod_{j=1}^n \delta (x_j)$$ but also cautions However, despite widespread use in engineering contexts, [it] ...
Galen's user avatar
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1 answer
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Domination for (finite) product measures equivalent to domination for marginals?

Let $\Omega=\Omega_1\times\Omega_2$ be a set equipped with a partial ordering that is inherited from partial orderings on $\Omega_1\times\Omega_2$. I.e. $(\omega_1,\omega_2)=\omega\leq\phi=(\phi_1,\...
jdods's user avatar
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3 votes
1 answer
134 views

Radon-Nikodym derivative with respect to product of marginal measures

Let $\mu$ be a (finite if necessary) measure on the product $\sigma$-algebra $\mathcal A_1 \otimes \mathcal A_2$ of two measurable spaces $(\Sigma_1,\mathcal A_1)$, $(\Sigma_2, \mathcal A_2)$. The ...
Michael's user avatar
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1 answer
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Do the finite product probability measures converge to the infinite product probability measure?

Is the limit of the finite product probability measures equal to the infinite product probability measure?\ $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\A}{\mathfrak{A}}$ To be precise: Let $(P_n)_{n\...
Matthias Georg Mayer's user avatar
8 votes
1 answer
130 views

Lipschitzness of $f$ on a high-probabilty subset from $\|f(X)-f(Y)\|\le \|X-Y\|$ with high probability for iid $(X,Y)$

Consider two iid random vectors $(X,Y)$ both in $R^k$ and $f:R^k\to R^m$. Assume that for some $\epsilon>0$ (that may be assumed small if necessary, say, $\epsilon<0.01$), it holds that $$P \Big(...
jlewk's user avatar
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1 vote
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Product of Lebesgue Measure is Lebesgue Measure of Products

I would like to show the Lebesgue measure $\lambda_{n + m}$ in $\mathbf{R}^{n + m}$ has the product property $$ \lambda_{n + m}(A \times B) = \lambda_n(A) \cdot \lambda_m(B) $$ for $A \in \mathcal{B}_{...
Mathematics_Beginner's user avatar
2 votes
0 answers
22 views

Construction of Higher Dimensional Product Measure: Directly and Inductively

To construct a product measure space generated by $(X, \mathcal{F}, \mu)$ and $(Y, \mathcal{G}, \lambda)$, we start with an algebra consisted of finite union of the generalized rectangles $A \times B \...
Mathematics_Beginner's user avatar
3 votes
1 answer
105 views

What is the "Natural" Product Measure on $\mathbf{R}^n$?

It seems like most of the times when we talk about product measures $\mathbf{R}^n$, we are talking about Lebesgue measure on $\mathbf{R}^n$. I will denote this measure space as $(\mathbf{R}^n, \...
Mathematics_Beginner's user avatar
1 vote
0 answers
34 views

How to prove the following equality in product measure

I was doing a theorem and I got stuck in this part. Let us $(\Omega_1,S_1,\mu_1)$ and $(\Omega_2,S_2,\mu_2)$ be sigma-finite measure space i.e $\exists A_n\in F_1(S_1\times S_2)$ such that $A_n\...
Andyale's user avatar
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0 answers
46 views

Question about product measure

Let me define first: let us have $(\Omega_1,S_1,\mu_1)$ and $(\Omega_2,S_2,\mu_2)$ be measure spaces. If $E\subseteq \Omega_1\times\Omega_2$ then the $x$-slice of $E$ is defined by $E^x=\{y\in\Omega_2:...
Andyale's user avatar
  • 107
1 vote
1 answer
80 views

Integrating with respect to marginal distribution

Let $(\Omega,\mathcal F, P)$ be a probability space and let $X$ and $Y$ denote continuous $\mathbb R^d$-valued random variables. The pushforward measures of $P$ under $X$ and $Y$ are denoted by $P_X$ ...
Syd Amerikaner's user avatar
2 votes
1 answer
84 views

Little confusion about Lebesgue's integral notation

So, I'm having a bit of trouble interpreting the integral of a integrable function over a measurable set. Let's say we have a measure space $(X,\,\Sigma,\,\mu)$, $A \in \Sigma$ a measurable set and $...
Fingolfin's user avatar
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0 answers
59 views

Motivation behind definition of product measure.

Consider two $\sigma$-finite measure spaces $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$.In standard measure theory books they define product measure as follows: $(\mu\times \nu)(E)=\int_X\nu(E_x)d\...
Kishalay Sarkar's user avatar
2 votes
0 answers
29 views

Measurability and equality of functions on product measure space

Let $(\Omega_1,\mathcal{A}_1,\mu_1)$, $(\Omega_2,\mathcal{A}_2,\mu_2)$ be two finite measure spaces and denote by $(\Omega_1 \times \Omega_2,\mathcal{A}_1 \otimes \mathcal{A}_2,\mu_1 \otimes \mu_2)$ ...
Matthis Stresemann's user avatar
0 votes
1 answer
70 views

How do I prove this statement about density in $[0,1]$?

I have the following problem: Let $\Omega =[0,1]$ the closed unit interval endowed with the standart topology and let $A\subset \Omega$. We denote by $\nu$ the outer lebesgue measure on $\Omega$. We ...
user123234's user avatar
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6 votes
2 answers
350 views

Folland: Why is the product measure well-defined?

Consider the following fragment from Folland's "Real analysis" on p64: I don't understand why $\pi(E)$ is well-defined, i.e. assume that $$E = \bigcup_i A_i \times B_i= \bigcup_j C_j \times ...
user avatar
0 votes
1 answer
15 views

Inequality between $L^2$ norms

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be two nonempty opens. Let $f \in L^2(X \times Y)$ and, for $x\in X$ fixed, $g_x(y)=f(x,y)$. Is that true that we always have $\lVert g_x \...
acd3456's user avatar
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0 answers
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Why is a rectangle $A \times B$ with $A \subset \mathcal{A}$, $B \in \mathcal{B}$ called a rectangle of $\mathcal{A} \times \mathcal{B}$?

In Freidman's Foundations of Modern Analysis, ch. 2.15 Product of Measures, he defines: The Cartesian product $X \times Y$ as all the ordered pairs $(x,y)$, where $x \in X$ and $y \in Y$. Rectangles ...
TOMILO87's user avatar
  • 510
2 votes
1 answer
159 views

Fibered product of probability spaces

Let $(\Omega_1, \mathcal{F}_1, \mu_1)$, $(\Omega_2, \mathcal{F}_2, \mu_2)$ and $(T, \mathcal{G}, \nu)$ be probability spaces. A probability space morphism is a measurable function that preserves the ...
Kristóf Marussy's user avatar
3 votes
1 answer
169 views

Show the left shift $T$ and $T^{-1} $ are measurable.

Let $X=\{0,1 \}^{\mathbb{Z}}$. So an element of $X$ is given by a sequence $(x_i)_{i\in \mathbb{Z}}$, where $x_i \in \{0,1 \}$. Let $\mu_i$ be a probability measure on $(\{0,1 \}, \mathcal{P}(\{0,1 \})...
Toasted_Brain's user avatar
2 votes
1 answer
138 views

Locally finite Borel measure on $\mathbb{R}^2$ that is not a product measure.

In the construction of Lebesgue-Stieltjes measures on $\mathbb{R}$, I have learned that a Borel measure that is finite on bounded intervals corresponds to a right-continuous increasing real-valued ...
Taxxi's user avatar
  • 1,502
0 votes
1 answer
169 views

Atomless?: product measure of an atomless measure and a measure which has atoms

I wonder whether the product measure of an atomless measure and a measure which has some atoms is atomless. Let us recall that given a measure space $(X, \mathscr A, \mu)$, the set $A \in \mathscr A$ ...
keisuke's user avatar
  • 75
1 vote
1 answer
176 views

Show that $f$ is measurable with respect to the product measure $c \times c$.

I am really stuck on this question. I am new in real analysis but interested. yet, I am not familiar with this. Let $(X, \mathcal{A},\mu) = (Y, \mathcal{B}, v) = (\mathbb{N}, \mathcal{M}, c)$. Here, ...
Maria Alora's user avatar
1 vote
0 answers
56 views

How do you show that the function $f(x)=\nu(P_x)$ is measurable?

There is a pretty classic question that is: Let $X=Y=[0,1]$, $\mathcal{M}=\mathscr{B}_{[0,1]}$, $\mathcal{N}=2^{[0,1]}$, let $\mu$ be Lebesgue measure on $\mathcal{M}$ and let $\nu$ be counting ...
Austin Jacobs's user avatar
0 votes
2 answers
379 views

Show there exists a set $E \subset \mathbb{R}^2$ such that the cross sections are open subsets but $E \notin \mathcal{B}_2$.

Show there exists a set $E \subset \mathbb{R}^2$ such that the cross sections $[E]_a$ and $[E]^a$ are open subsets for every $a \in \mathbb{R}$ but $E \notin \mathcal{B}_2$. This is a question from ...
Evan Kim's user avatar
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4 votes
1 answer
372 views

$f: X \rightarrow \mathbb{R}$ is measurable iff $\{(x, t): t=f(x)\} \in \mathcal{F} \otimes \mathcal{B}(\mathbb{R})$.

A function $f: X \rightarrow \mathbb{R}$ is measurable if and only if its graph is measurable, $\{(x, t): t=f(x)\} \in \mathcal{F} \otimes \mathcal{B}(\mathbb{R})$. This is easy when we consider the ...
XiaoMem24's user avatar
1 vote
1 answer
95 views

$K=\{E|f^{-1}(E)\in \mathcal{S} \}$ is a $\sigma$ algebra [closed]

This is a problem I have seen while self studying measure theory: Let $(X,\mathcal{S})$ be a measure space, and $E\in \mathcal{S} \otimes \mathcal{S}$, with $f(x) = (x,x)$ then $K=\{E|f^{-1}(E)\in \...
Jenny Liu's user avatar
  • 217
0 votes
0 answers
58 views

Getting Borel-measurability from a product measure space

I have that the graph of a function $G(f) = \{ (x, f(x)) : x \in \mathbb{R} \} $ for $f : \mathbb{R} → \mathbb{R} $ is such that $G(f) ∈ B \otimes B$. Here $B$ is the $\sigma$-algebra of Borel sets. ...
jonnywhite's user avatar
0 votes
2 answers
64 views

Finding $E \in \mathcal A \otimes \mathcal B$ such that $E \neq E^y \times E_x,$ for some $x \in X, y \in Y.$

Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be two measure spaces. What has been stated in my book is that $\mathcal A \times \mathcal B$ may not necessarily be a $\sigma$-algebra of subsets ...
Anil Bagchi.'s user avatar
  • 2,912
3 votes
0 answers
54 views

A question of why these integrands can be taken out of the integrals.

Rudin says that (2) can be rewritten as (4), as seen below. However, both integrands in (2) respectively depend on $x$ and $y$, which implies that (2) could not be rewritten as (4). Is there any ...
metric's user avatar
  • 513
1 vote
2 answers
117 views

Random variable greater than another random variable - measure theoretic argument?

Suppose $\xi,\eta$ are two independent, identical random variables that are non-degenerate. I want to show that $\mathbb{P}(\xi<\eta) = \mathbb{P}(\eta<\xi) > 0$ through a measure-theoretic ...
varpi's user avatar
  • 607
1 vote
0 answers
30 views

Show that there exists a unique measure $\pi$ on $\mathcal{B}[0,\infty)\otimes\mathcal{P}(\mathbb{N})$

I have already shown that $\Gamma\in \mathcal{B}[0,\infty)\otimes\mathcal{P}(\mathbb{N})$ iff $\Gamma = \bigcup_{j\in\mathbb{N}} A_j\times \{j\}$ where $A_j \in\mathcal{B}[0,\infty)$ for all $j$. I am ...
Andrew Shedlock's user avatar
1 vote
0 answers
72 views

Confusions with Fubini’s Theorem

Fubini's Theorem: Let $\mu,\nu$ be two $\sigma$-finite measures and $f$ be a measurable function. If $f \ge 0$ or $f \in L^1(\mu\times \nu)$, then $$ \int f ~d \mu \times \nu = \iint f ~d\mu ~d\nu = \...
wzstrong's user avatar
  • 553
0 votes
0 answers
216 views

Area under a curve proof using product measure

Let $u:\mathbb{R}\to[0,\infty]$ be a Borel measurable function, define the set $S[u] = \{(x,y):0\leq y\leq u(x)\}$ and let $\lambda^n$ be the Lebesgue measure for $\mathbb{R}^n$. I am trying to show ...
Andrew Shedlock's user avatar
1 vote
1 answer
157 views

Extending the definition of stochastic integral from simple processes

I am reading stochastic integration from Brownian Motion And Stochastic Calculus by Karatzas and Shreve. In the course of extending the definition of the stochastic integral from simple processes to ...
Sudheesh Surendranath's user avatar
0 votes
1 answer
431 views

Product measure of Lebesgue and counting measure on discrete and usual topology on $\mathbb{R}$

$\textbf{Corollary}$: Let $X,Y$ be locally compact Hausdorff spaces. Let $\mu$ and $\nu$ be regular Borel measures on $X$ and $Y$ respectively. If $E$ is a Borel subset of $X\times Y$ that is included ...
Schach21's user avatar
  • 700
1 vote
1 answer
108 views

Determine if $A$ and $B$ are independent (Product Measure, Independence)

Consider $\Omega_1 = \mathbb{N}_0$, $\Omega_2 = \mathbb{R}$, $\mathbb{P}_1 = Poi(5)$ and $\mathbb{P}_2 = Normal(1,2)$. On $\Omega = \Omega_1 \times \Omega_2$ we use the the product measure $\mathbb{P}$...
user avatar
2 votes
1 answer
125 views

An application of Fubini's theorem I don't understand

Here is a result asserting that, under some technical conditions, we can replace a Borel almost everywhere homomorphism by a Borel homomorphism. It is taken from Zimmer's book "Ergodic Theory and ...
LBJFS's user avatar
  • 1,345
1 vote
2 answers
305 views

Measure Theory: $X$, $Y$ independent, proof that $P_{X+Y} = P_{X} * P_{Y}$

I am studying measure theory, and I bumped in the the following: Let $X$ and $Y$ be two independent random variables with distributions $P_X$ and $P_Y$ respectively. A lot of sources conclude (or use ...
hello's user avatar
  • 207
1 vote
1 answer
131 views

Which elements are contained in an infinite product space?

I have a question about the formal treatment of infinite product measures in probability calculus. Take for example the model of an infinite coin toss. If $(\Omega_i,\mathcal A_i,P_i)$ is the ...
Nils Hoppenstedt's user avatar
1 vote
2 answers
59 views

Joint Random Variables

I am trying to clarify the meaning of a joint pdf. A collection of random variables $\{X_i\}_{i=1}^{n}$ that are i.i.d. defined on the space $(\Omega, F, P)$, we define their joint probability as: ...
nvm's user avatar
  • 1,306
1 vote
1 answer
45 views

Equality of certain integrals over product space

Let $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ be measurable spaces. (I'm happy to restrict this to standard Borel spaces.) Suppose there are finite measures $\mu$ and $\nu$ on $X$ and a finite kernel $k$ ...
daon's user avatar
  • 339