exercise_on_probability_essentials_chp.2-lastcbj

exercise_on_probability_essentials_chp.2-lastcbj - Printed...

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Unformatted text preview: Printed April 10, 2009 Exercise on Probability Essentials By Author. Jean Jacod, Philip Protter Byoung jin Choi e-mail: [email protected] Abstract In this book we solve a exercise of the Probability Essentials wrote by Jean Jacod, Philip Protter. Author thanks to graduate student of Department of Mathematics in Chungbuk National University in Republic of Korea. 1 1 Introduction 2 Axioms of probability Exercise 2.2 Let ( G α ) α ∈ A be an arbitrary family of σ-algebras defined on an abstract space Ω. Show that H = ∩ α ∈ A G α is also a σ-algebra. Proof. We check H = ∩ α ∈ A G α has the three properties of a σ-algebra (1) since for any α ∈ A , G α is σ-algebras, therefore Ω , ∅ ∈ ∩ α ∈ A G α (2) If A ∈ ∩ α ∈ A G α then A ∈ G α for any α ∈ A . Therefore A c ∈ G α for any α ∈ A . So A c ∈ ∩ α ∈ A G α (3) Let ( A n ) be a sequence in H . Since for each A n ∈ G α , ∪ ∞ n =1 A n ∈ G α . So ∪ ∞ n =1 A n ∈ ∩ α...
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exercise_on_probability_essentials_chp.2-lastcbj - Printed...

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