Is there a conditional random variable?

Question

Let $(\Omega,\Sigma,\mu)$ be a sample space.

Let $F $ be a $\sigma$-subalgebra of $\Sigma$ and let $X$ be a real-valued random variable.

So what does it mean $X/F$? How is this defined mathematically and books to read?

Answer 1

Let $X$ be a R.V. on $(\Omega, \mathcal{F}_0, P)$ such that $E|X| < \infty$.

$E[X \mid \mathcal{F}]$ where $\mathcal{F}$ a $\sigma$-algebra such that $\mathcal{F} \subset \mathcal{F}_0 $is any random variable $Y$ (called a ``version'' of $E[X \mid \mathcal{F}]$) that satisfies:

1) $Y$ is $\mathcal{F}$-measurable

2) for all $A \in \mathcal{F}$, $\int_A X \, dP = \int_A Y \, dP$

It exists and is unique.

A good book to read is Durrett's Probability: Theory and Examples (now in its 4th edition but a 2nd or 3rd edition is fine) - this definition is taken from the 3rd edition, section 4.1.