Constant invariant quantities

What is the intuition behid the constant invariant quantities and why do we need it? How is this related with probability theory?

I was solving some maximization problem and when I saw the answer, the author mendtioned that since the quantity $mathbf{B}in L^2(Omega,mathbb{F},mathbb{P})$ is constant invariant then it holds that $mathbb{E}(mathbf{B})=0$ and she only considers – uses the variance of the r.v. to find the maximum.