[Math] Generalized marginal probability and marginal expectation

measure-theoryprobability theory

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?

Related Question