handoutmpSP06 - A Primer on Markov Processes Markov...

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APr imeronMarkovProcesses Markov processes will prove useful to represent uncertainty in dynamic pro- gramming problems. See Ch. 7,8,11 of S-L. Def. Let S be a set and let S be a collection of subsets of S . S is called a σ algebra if: (1) ,S S ;(2) A S implies A c = S \ A S (closed under complementation); and (3) A n S implies n =1 A n S (closed under countable union) By DeMorgan’s law it is also closed under intersection. Ex. S = { Heads,Tails } . Then the power set of S is { , { H } , { T } ,S } . Notice this satis f es all the properties of an algebra. Def. ( S, S ) is called a measurable space if S is a σ -algebra of S . Def. Let ( S, S ) be a measurable space. A measure is a real valued function µ : S R such that: (1) µ ( )=0 ;(2) µ ( A ) 0 for all A S ; and (3) if { A n } is a countable, disjoint sequence of subsets in S ,th en µ ( A n )= P µ ( A n ) . Def. If µ ( S )=1 ,then µ is a probability measure. Ex. µ ( H )= 1 2 . Def. Given a measurable space ( S, S ), a real-valued function f : S R is ( S ) measurable if { s S : f ( s ) a } S for all
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handoutmpSP06 - A Primer on Markov Processes Markov...

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