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Understanding Infinite Probability Spaces and Sigma-Algebras in Probability Theory

This chapter explores the complexities of infinite probability spaces, detailing the distinctions between finite and infinite sets, including countable and uncountable sets. It introduces the concept of sigma-algebras, defining their essential properties. The chapter also discusses practical examples, such as the communication challenges between a colorblind individual and an illiterate person in forming a shared information set. By using Cantor’s diagonal argument, readers gain insight into the nature of uncountable sets and their significance in probability theory.

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Understanding Infinite Probability Spaces and Sigma-Algebras in Probability Theory

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  1. Chapter 1 General Probability Theory

  2. 1.1 Infinite Probability Spaces • Finite • Infinite • Counterable (enumerate) • Uncounterable (real、無理數)

  3. Cardinality

  4. Cantor’s Diagonal Argument S1 = (1、1、0、………) S2 = (1、0、0、………) S3 = (0、1、0、………) . . . = ( ) 找一個S‘ = (0、1、0、………) 不屬於 的序列，但是為

  5. Definition 1.1.1 • Let  be a nonempty set and let F be a collection of subsets of  . We say that F is a -algebra provided that: (1) the empty set  belongs to F (2) whenever a set A belongs to F, its complement also belongs to F (3) whenever a sequence of sets belongs to F, their union also belongs to F

  6. Definition 1.1.2

  7. Note

  8. -algebra:資訊集合 • 假定有一個色盲,不能分辨顏色,只能分辨“正”“反” • “正” 事件 {“紅正”,“綠正”} • “反” 事件 {“紅反”,“綠反”} • 色盲的資訊集合F1={,, {“紅正”,“綠正”}, {“紅反”,“綠反”}} • 假定有一個文盲,不能分辨文字,只能分辨“紅”“綠” • “紅” 事件 {“紅正”,“紅反”} • “綠” 事件 {“綠正”,“綠反”} • 文盲的資訊集合F2={,, {“紅正”,“紅反”}, {“綠正”,“綠反”}}

  9. -algebra:資訊集合 • 當色盲和文盲溝通,就可透過推論辨別文字和顏色 • Ex: {“紅正”,“綠正”} {“紅正”,“紅反”}={“紅正”} • {“紅正”,“綠正”} {“紅正”,“紅反”}={“紅正”,“綠正”, “紅反”} “綠反”事件未發生

  10. -algebra:資訊集合 • 文盲和色盲溝通後建立新的資訊集合F • F= {,, {“紅正”,“綠正”}, {“紅反”,“綠反”}, {“紅正”,“紅反”}, {“綠正”,“綠反”}, {“紅正”},{“綠正”},{“紅反”},{“綠反”} {“紅正”,“綠反”}, {“綠正”,“紅反”}, {“紅正”,“綠正”,“紅反”},{“紅正”,“綠正”,“綠反”}, {“紅正”,“紅反”,“綠反”},{“紅反”,“綠正”,“綠反”}, } 兩個人的資訊集合 共 元素 再做推論

  11. Example 1.1.3 (Uniform (Lebesgue) measure on [0,1])

  12. Definition 1.1.5

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