# Chap2 - Chapter 2 Information and Conditioning Definition...

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Unformatted text preview: Chapter 2. Information and Conditioning Definition 2.1.1. Let Ω be a nonempty set. Let T be a fixed positive number, and assume that for each t ∈ [0 , T ] there is a σ-field F ( t ). Assume further that if s ≤ t , then every set in F ( s ) is also in F ( t ). Then we call the collection of σ-fields F ( t ) , ≤ t ≤ T , a filtration. Definition 2.1.3. Let X be a random variable defined on a nonempty sample space Ω. The σ- field generated by X , denoted σ ( X ), is the collection of all subsets of Ω of the form { X ∈ B } , where B ranges over the Borel subsets of R . Definition 2.1.5. Let X be a random variable defined on a nonempty sample space Ω. Let G be a σ-field of subsets of Ω. If every set in σ ( X ) is also in G , we say that X is G-measurable. 1 Definition 2.1.6. Let Ω be a nonempty sam- ple space equipped with a filtration F ( t ) , ≤ t ≤ T . Let X ( t ) be a collection of random variables indexed by t ∈ [0 , T ]. We say this collection of random variables is an adapted stochastic process if, for each t , the random variable X ( t ) is F ( t )-measurable....
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Chap2 - Chapter 2 Information and Conditioning Definition...

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