Skip to main content

Questions tagged [product-measure]

The tag has no usage guidance.

92 questions
Filter by
Sorted by
Tagged with
1 vote
2 answers
35 views

• 904
1 vote
0 answers
32 views

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 ...
• 41
2 votes
2 answers
126 views

• 8,097
3 votes
1 answer
85 views

• 647
1 vote
1 answer
43 views

• 6,330
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 ...
• 365
0 votes
1 answer
49 views

• 616
0 votes
0 answers
45 views

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 ...
• 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 ...
3 votes
1 answer
169 views

• 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. ...
• 105
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 ...
• 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 ...
• 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 ...
• 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 ...
• 2,577
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 = \...
• 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 ...
• 2,577
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 ...
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 ...
• 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 $\mathbb{P}$...
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 ...
• 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 ...
• 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 ...
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: ...
• 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 $k$ ...
• 339