Conditional Expectation
نویسنده
چکیده
Let μ and λ be two positive bounded measures on the same meaurable space (Ω,F). We call μ and λ equivalent, and write μ ≡ λ, if they have the same null sets— so, if they were probability measures, the notion of “a.s.” would be the same for both. More generally, we call λ absolutely continuous (AC) w.r.t. μ, and write λ μ, if μ(A) = 0 implies λ(A) = 0, i.e., if every μ-null set is also λ-null. We call μ and λ mutually singular, and write μ ⊥ λ, if for some set A ∈ F we have μ(Ac) = 0 and λ(A) = 0, so μ and λ are “concentrated” on disjoint sets. For example— if λ(A) = ∫
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