# Questions tagged [product-measure]

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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Rick Durrett, Probabilty Theory and Examples, Lemma 2.2.8

I have a question from Rick Durrett's Probability Theory and Examples. In that book, Lemma 2.2.8 If $Y \ge 0$ and $p > 0$, then $E(Y^p) = \int^{\infty}_{0} py^{p-1}P(Y>y)\,dy$ Proof. Using ...
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### Extending Fubini's Theorem to general joint distributions

Take Fubini's theorem as stated above. Say there exists $(M,\mathscr{M}, \mu)$ and $(N,\mathscr{N},\nu)$, then the product measure is such that $\pi(A\cap B)=\mu(A)\nu(B)$ where $\pi$ is defined over ...
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### Calculating a product measure with Fubini-Tonelli's theorem

I am kind of lost on the following problem. Let $${f : (X, \mathcal{A}, \mu) \rightarrow (\mathbb{R_{\geq}}, \mathcal{B}(\mathbb{R_{\geq}}), \lambda)}$$ with ${\lambda}$ being the Lebesgue-measure. 1)...
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### Calculate product measure of off-diagonal set

Let $(E,\mathcal E)$ be a measurable space $\mu_i$ be a probability measure on $(E,\mathcal E)$ $\lambda:=\mu_1+\mu_2$ and $$f_i:=\frac{{\rm d}\mu_i}{{\rm d}\lambda}$$ $Q$ denote the measure with ...
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### Strong convergence in $L^1$, $\mathbb{R}^d$ with $d>1$

A sequence of positive function $f_n(x)\rightarrow f(x)$ pointwise on the unit ball $B:=\{x:|x|\leq 1\}\subset \mathbb{R}^d$ as $n\rightarrow \infty$. Suppose the sets $\Lambda_t^n=\{x:f_n(x)\geq t\}$ ...
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### Prove $\{(\omega,y) \in \Omega \times [0,\infty) \ | \ y \leqslant f(\omega) \}$ is in the product $\sigma$-algebra $\mathcal{A} \otimes \mathcal{B}$

We have a $\sigma$-finite measure space $(\Omega,\mathcal{A},\mu)$, a measurable nonnegative function $f:\Omega \to \mathbb{R}$ and \begin{equation} G_f := \{(\omega,y) \in \Omega \times [0,\...
### double integral of f zero over subsets of $[0,1]^2$ implies $f=0$ almost everywhere
Here's the statement of the problem: Let f $\epsilon L^1([0,1]^2,\mathscr{B}(\mathbb{[0,1]^2}),m)$, where $\mathscr{B}(\mathbb{[0,1]^2})$ is the Borel $\sigma$-algebra and m is the two-dimensional ...
Let $f: \left [ 0, 1 \right ] \times \left [ 0, 1 \right ] \rightarrow \mathbb{R}$ satisfy: $f_x\left ( y \right ):= f\left ( x, y \right ) : \left [ 0, 1 \right ]\rightarrow \mathbb{R}$ is Riemann ...