First, to be clear, there are many questions about how the power set is a $\sigma$-algebra and this is not one of those :)
My question is: ''Does there exists a natural choice of $\sigma$-algebra on the power set $P(X)$ for a measurable space $(X,\Sigma)$?''
Similar to the case of topological spaces, I assume it will be necessary to put certain requirements on $X$ to get an affirmative answer? (E.g. standard Borel/Polish).