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Constant invariant quantities

Mathematics Asked on November 6, 2021

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.

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