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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