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Let R be the set of real numbers. Say we have a function $f(x,X) \in R $ where $X \in R^d$ is a random variable over $\Omega$ and $x \in R$. I'm searching for an upper bound for the expected value of
$$E(\max_{x \in R} |f(x,X) - E(f(x,X)|)$$
under some lighter constraints like $f$ is bounded.
Any help is appreciated!
A really awful estimate is to say that $|f-E|\leq |f| + |E|$ and so $\max_x |f(x)-E| \leq \max_x(|f|+|E|) = M + |E|$ where $M = \sup_x |f(x)|$, which is finite as your assumption is that $f$ is bounded. Then $\int_{\Omega} \max_x |f(x)-E| ~ d\mu(x) \leq (M+|E|)\mu(\Omega)$. If we know that $\mu(\Omega) = 1$ then we can bound your integral by this constant. But I am not sure how good this is for what you want.
