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Hi, I'm looking for conditions on $G(t,x)$ such that $$\sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)]$$ where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [0,1]}E[G(t,X)]\leq E[\sup\limits_{t\in [0,1]}G(t,X)]$).

Any suggestion or reference is greatly appreciated!

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Hi, I'm looking for conditions on $G(t,x)$ such that $$\sup\limits_{t\in I}E[G(t,X)]=E[\sup\limits_{t\in I}G(t,X)] [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)] where  I=[0,1] and X is a random variable (it's easy to see that \sup\limits_{t\in I}E[G(t,X)]\leq [0,1]}E[G(t,X)]\leq E[\sup\limits_{t\in I}G(t,X)]).[0,1]}G(t,X)]). Any suggestion or reference is greatly appreciated! 2 deleted 12 characters in body Hi, I'm looking for conditions on G(t,x) such that$$ \sup\limits_{t\in I}E[G(t,X)]=E[\sup\limits_{t\in I}G(t,X)]  where $I\subset \mathbb{R}$ I=[0,1]$and$X$is a random variable (it's easy to see that$\sup\limits_{t\in I}E[G(t,X)]\leq E[\sup\limits_{t\in I}G(t,X)]\$).

Any suggestion or reference is greatly appreciated!

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