My question is about the conditional expectation of random variables with respect to a $\sigma$-algebra. I am having trouble getting an intuition behind the definitions among other things.
I know that if $X$ is $\mathcal{G}$-measurable then $\mathbb{E} [X| \mathcal{G}] = X$, but what if $X$ is not $\mathcal{G}$-measurable? Is this expression just not defined?
Furthermore, in the following definition:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $\mathcal{G}$ a sub $\sigma$-field of $\mathcal{F}$ and $X$ be a random variable with $\mathbb{E}|X| < \infty$.
Definition (Conditional Expectation): The conditional expectation of $X$ given $G$, denoted by $\mathbb{E}[X|\mathcal{G}]$, is defined as any random variable Y which satisfies
- $Y$ is $\mathcal{G}$-measurable
- $\int_A X d\mathbb{P} = \int_A Y d\mathbb{P}$ for all $A \in \mathcal{G}$
Remark:
- Any random variable $Y$ satisfying $(1)$ and $(2)$ in the definition is called a version of $\mathbb{E}[X|\mathcal{G}]$.
I don't know why the 2nd condition is defined this way. Can someone bring an argument where if the second condition is not defined in the way it is then something undesirable results? Or an argument where this 2nd condition brings desirable results.