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Currently, I am working on some sort of stochastic optimization problems defined over function spaces.

I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples"). However, the standard discussions are restricted to random objects in Euclidean spaces (random vectors and variables). I like to know whether there are some proper references on extensions of these notions and theorems (WLLN, SLLN, CLT, Conditional Expectation,...) to the case where the objects of interest belong to a function space.

Thanks

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You can start with "Analysis and Probability on Infinite-Dimensional Spaces" by Nathaniel Eldredge, and the references therein: http://arxiv.org/abs/1607.03591

For a comprehensive treatment of probability in Banach spaces, see Ledoux and Talagrand: "Probability in Banach Spaces", Springer 2011.

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