Conditional variance of power set

Question

Conditional Variance is expressed as

Var(X|Y) = Var(X|$\sigma$(Y)).

And unconditional variance

Var(X) = Var(X|$\mathcal{F_0}$)

where $\mathcal{F_0}$ is the trivial information set.

Then is Var(X|$\mathcal{P(\Omega)}$)= $0$

regardless of how I put X as a random variable ?

Answer 1

Any random variable $X$ is measurable w.r.t. the power set. Hence $E(X^{2}|\mathcal P(\Omega))=X^{2}$ and $E(X|\mathcal P(\Omega))=X$ so the conditional variance is $X^{2}-X^{2}=0$. Intuitively any random variable is a constant when conditioned on the power set and constants have variance $0$.