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## A.7 Some functional analysis

Numerous arguments in this book involve infinite-dimensional vector spaces. Typical examples arising in connection with a probability space (Ω,F, P) are the spaces Lp := Lp(Ω,F, P) for 0 p , which we will introduce below. To this end, we first take p (0,] and denote by Lp(Ω,F, P) the set of all F-measurable functions Z on (Ω,F, P) such that Zp < , where

Let us also introduce the space L0(Ω,F, P), defined as the set of all P-a.s. finite random variables. If no ambiguity with respect to σ-algebra and measure can arise, we may sometimes write Lp(P) or just Lp instead of Lp(Ω,F, P). For p [0,], the space

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