In the Gaussian case, it is well-known that the MSE, is minimizer by the mean value. However, in general, if $X \in L^2(\mathcal{F};\mathbb{P})$, is a random-variable in $\mathbb{R}$, then is the does $$ \mathbb{E}_{\mathbb{P}}[X] = \inf_{Z \in L^2(\mathcal{F};\mathbb{P})} \;\mathbb{E}_{\mathbb{P}}\left[(X-Z)^2\right]? $$
The argmin, is the conditional expectation, but what about the min itself?