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Now, Im = A ∩ Im = n In Im (f) m m (g) = µ(A). 1. ✷ CHAPTER 1. 1 measures. 3 (Probability distributions P (·) and dfs F (·)) (a) In probability theory we think of Ω as the set of all possible outcomes of some experiment, and we refer to it as the sample space. The individual points ω in Ω are referred to as the elementary outcomes. The measurable subsets A in the collection A are referred to as events. A measure of interest is now denoted by P ; it is called a probability measure, and must satisfy P (Ω) = 1.
E. 3 Consider a measure space (Ω, A, µ). Let µ0 ≡ µ|A0 for a sub σ-ﬁeld A0 of A. Starting with indicator functions, show that X dµ = X dµ0 for any A0 -measurable function X. CHAPTER 3. INTEGRATION 44 3 Evaluating and Diﬀerentiating Integrals Let (R, Bˆµ , µ) denote a Lebesgue–Stieltjes measure space that has been completed. If g is Bˆµ -measurable, then g dµ is called the Lebesgue–Stieltjes integral of g; and if F is the generalized df corresponding to µ, then we also use the notation g dF . b Also, a g dF ≡ (a,b ] g dF = 1(a,b ] g dF .
A. µ2 (AΩn ) since µ1 = µ2 on σ[C ∩ Ωn ] = σ[C] ∩ Ωn , by claim 7 (m) = µ2 (A) completing the proof. , ✷ Question We extended our measure µ from the ﬁeld C to a collection A∗ that is at least as big as the σ-ﬁeld σ[C]. Have we actually gone beyond σ[C]? Can we go further? 2. 2 (Complete measures) Let (Ω, A, µ) denote a measure space. If µ(A) = 0, then A is called a null set. We call (Ω, A, µ) complete if whenever we have B ⊂ (some A) ∈ A with µ(A) = 0, we necessarily also have B ∈ A. 1 (Completion) (4) Let (Ω, A, µ) denote a measure space.