It seems that the derivations of the maximum entropy distributions is a "well-known" fact and so it is in the continuous and discrete cases.... However, I can't seem to find a proof/formal statement in the general case on a general probability space $(\Omega,\mathcal{F},\mathbb{P})$?

The proof in the continuous case can be found in Theorem 12.1.1 here.. but what about the general case for an arbitrary Borel reference measure?