# Generalized marginal probability and marginal expectation

I think, that I need something like the following, but do not find it anywhere in textbooks. I am not even sure if it makes sense.

If you recognize it, please provvide some pointers.

• Two measurable spaces $(\Omega_i,\mathcal{F}_i)$, with $i\in\{1,2\}$
• A product space with probability $(\Omega=\Omega_1\times\Omega_2,\mathcal{F}=\mathcal{F}_1\times\mathcal{F}_2,P)$
• Projection(better name?): $\pi_1(A\in\mathcal{F}_1):=A\times \Omega_2$ and $\pi_2(A\in\mathcal{F}_2):=\Omega_1 \times A$
• Also possible: $\pi_1(\mathcal{F_1})=\{A\times\Omega_2|A\in\mathcal{F}_1\}$

Now you can have something like marginal probabilities? $P_i$ is a probability measure on $\mathcal{F}_i$:

$P_i(A):=P(\pi_i(A))$

In the classical two random variable setting, one has expectation conditional on one of the two. I would generalize this as conditional expectation with respect to the sigma-algebra $\pi_i(\mathcal{F}_i)$, e.g.

$E[X|(\pi_i\mathcal{F}_i)]$

This should integrate out any dependence on states of the $i$-th space?

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a) If $\mathcal{F}_i$ is a sigma-algebra, should'nt $\pi_i(\mathcal{F}_i)\subset \mathcal{F}$ also be one? b) Where can I read about marginals of $P$? c) I can always take an expectation restricted on any set $\pi_1(A), A\in\mathcal{F}_1$. The set of all $\pi_1(A)$ is $\pi_1(\mathcal{F}_1)$, so I thought this could be captured by a conditional expectation. d) In this setting I do not have random variables, but in the two variable setting, there is a marginal expectation. I am looking for something similar for my situation. – JSG Jun 24 '12 at 21:35