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Extra resources for An Introduction to Probability Theory
In this case one has |❊f | ≤ ❊|f |. ❊f = 0. s. s. s. and ❊f exists, then ❊g exists and ❊f = ❊g. , then (4) (5) Proof. (1) follows directly from the definition. Property (2) can be seen as follows: by definition, the random variable f is integrable if and only if ❊f + < ∞ and ❊f − < ∞. Since ω ∈ Ω : f + (ω) = 0 ∩ ω ∈ Ω : f − (ω) = 0 = ∅ and since both sets are measurable, it follows that |f | = f + +f − is integrable if and only if f + and f − are integrable and that |❊f | = |❊f + − ❊f − | ≤ ❊f + + ❊f − = ❊|f |.
5. FUBINI’S THEOREM 51 for all ∞ < a < b < ∞ where p : ❘ → [0, ∞) is a continuous function such ∞ that −∞ p(x)dx = 1 using the Riemann-integral. 2. 1 and that Pϕ = ∞ pk δηk k=1 with pk ≥ 0, ∞ k=1 pk = 1, and some ηk ∈ E (that means that the image measure of P with respect to ϕ is ’discrete’). 5 g(ϕ(ω))dP(ω) = ❘ g(η)dPϕ (η) = ∞ pk g(ηk ). k=1 Fubini’s Theorem In this section we consider iterated integrals, as they appear very often in applications, and show in Fubini’s Theorem that integrals with respect to product measures can be written as iterated integrals and that one can change the order of integration in these iterated integrals.
Then ak = 0 implies P(Ak ) = 0. s. Properties (4) and (5) are exercises. The next lemma is useful later on. In this lemma we use, as an approximation for f , a staircase-function. 3. 2 Let (Ω, F, P) be a probability space and f : Ω → random variable. ❘ be a (1) Then there exists a sequence of measurable step-functions fn : Ω → such that, for all n = 1, 2, . . and for all ω ∈ Ω, |fn (ω)| ≤ |fn+1 (ω)| ≤ |f (ω)| and ❘ f (ω) = lim fn (ω). n→∞ If f (ω) ≥ 0 for all ω ∈ Ω, then one can arrange fn (ω) ≥ 0 for all ω ∈ Ω.