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Lec2_0924 - Last Time Introduction Measurable space...

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Last Time Introduction Measurable space Generated σ -fields Borel σ -field Today’s lecture: Sections 1.1–1.2.2 MATH136/STAT219 Lecture 2, September 24, 2008 – p. 1/14
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Probability space A probability space is a triple , F , IP ) , where , F ) is a measurable space and IP is a probability measure A probability measure is a set function IP : F → [0 , 1] which satisfies: IP (Ω) = 1 0 IP ( A ) 1 for all A ∈ F Countable additivity : if A i ∈ F , i = 1 , 2 , . . . are mutually disjoint (i.e. A i A j = , i negationslash = j ) then IP ( uniondisplay i =1 A i ) = summationdisplay i =1 IP ( A i ) If IP ( A ) = 1 then we say A occurs almost surely (a.s.) MATH136/STAT219 Lecture 2, September 24, 2008 – p. 2/14
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Specifying the Probability Measure IP Countable Ω Set F = 2 Ω Define p ω for each ω Ω , such that 0 p ω 1 and w Ω p w = 1 Then IP ( A ) . = ω A p w defines a probability measure on , 2 Ω ) Uncountable Ω Consider a set of generators { A α : α Γ } with F = σ ( { A α : α Γ } ) Define a probability measure IP ( A α ) for all A α , α Γ Then (under mild conditions) IP extends uniquely to a probability measure on , F ) (see Rosenthal, Section 2.3) MATH136/STAT219 Lecture 2, September 24, 2008 – p. 3/14
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Some properties of IP Let , F , IP ) be a probability space and A, B, A i , B i ∈ F , i = 1 , 2 , . . .
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