Consider a probability space $(\Omega,\mathcal{F},P)$ and some sub-$\sigma$-algebra $\mathcal{G}\subseteq\mathcal{F}$. Can the conditional distribution
$$P(A|\mathcal{G})=E_P[1_{A}|\mathcal{G}],\quad\text{for }A\in\mathcal{F}$$
be an actual probability measure on $\mathcal{G}$ and be used as such? For example for a random variable $X$, we would have
$$P(X\in A|\mathcal{G})=P(\cdot|\mathcal{G})(X^{-1}(A))\quad$$
and so forth?
Or let me ask in another way: Should I understand a conditional distribution as an hypothetical distribution?
Best Answer
The value $P(A|\mathcal G)$ is a $\mathcal G$-measurable function, not a number. So $A \mapsto P(A|\mathcal G)$ is not a measure.
You might attempt a "disintegration" where, for each $\omega \in \Omega$, the set-function $A \mapsto P(A|\mathcal G)(\omega)$ is a probability measure defined on $\mathcal F$. [Search "disintegration" to find conditions under which this can be done.]
Note, it is not very interesting on $\mathcal G$, since when $A \in \mathcal G$, we have $P(A|\mathcal G) = \mathbf1_{A}$.