A Theory of Probability Homework 1 Solutions
نویسنده
چکیده
Exercise [1.1.13] (a) Let G = ⋂ α Fα, with each Fα a σ-algebra. Since Fα a σ-algebra, we have that Ω ∈ Fα, and as this applies for all α, it follows that Ω ∈ G. Suppose now that A ∈ G. That is, A ∈ Fα for all α. Since each Fα is a σ-algebra, it follows that A ∈ Fα for all α, and hence A ∈ G. Similarly, let A = ⋃ iAi for some countable collection A1, A2, . . . of elements of G. By definition of G, necessarily Ai ∈ Fα for all i and every α. Since Fα is a σ-algebra, we deduce that A ∈ Fα, and as this applies for all α, it follows that A ∈ G.
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