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cond.meas-prob - Conditional probability conditional...

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Conditional probability & conditional measures Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 4 May, 2010 (at 23:20 ) Abstract: Abs. cty, Radon-Nikodym thm. Elementary martingale thy. In progress: As of 4May2010 Bonjour. As additional notation 1 use to mean identically equals ’; on the probability space, we mean this a.e. Use r.var or r.v for ‘ random variable ’. Use r.walk for ‘ random walk ’. Sets & Fields. Use for “is an element of”. Letting P be the set of primes, then, 5 P yet 6 / P . Changing the emphasis, P 3 5 ( P owns 5” ) yet P 63 6. For subsets A and B of the same space, Ω, the inclusion relation A B means: ω A , necessarily B 3 ω . And this can be written B A . Use A $ B for proper inclu- sion, i.e, A B yet A 6 = B . The difference set B r A is { ω B | ω / A } . Employ A c for the complement Ω r A . Use A 4 B for symmetric 1 Phrases: WLOG : ‘ Without loss of generality ’. TFAE : The following are equivalent ’. OTForm : ‘ of the form ’. FTSOC : For the sake of contradiction ’. Use iff : ‘ if and only if ’. IST : ‘ It Suffices to ’ as in ISTShow , ISTExhibit . Use w.r.t : ‘ with respect to ’ and s.t : ‘ such that ’. Latin: e.g : exempli gratia , ‘ for example ’. i.e : id est , ‘ that is ’. Number Sets: An expression such as k N ( read as “ k is an element of N ” or “ k in N ) means that k is a natural number; a natnum . N = natural numbers = { 0 , 1 , 2 , . . . } . Z = integers = { . . . , - 2 , - 1 , 0 , 1 , . . . } . For the set { 1 , 2 , 3 , . . . } of positive integers, the posints , use Z + . Use Z - for the negative integers, the negints . Q = rational numbers = { p q | p Z and q Z + } . Use Q + for the positive ratnums and Q - for the negative ratnums. R = reals. The posreals R + and the negreals R - . C = complex numbers, also called the complexes . Mathematical objects: Poly(s) : polynomial(s) ’. Irred : irreducible ’. Coeff : coefficient ’ and var(s) : variable(s) ’ and parm(s) : parameter(s) ’. Expr. : expression ’. Fnc : function (so ratfnc : means rational function , a ratio of polynomials). Seq : sequence ’. Soln : ‘ solution ’. Prop’n : ‘ proposition ’. CEX : ‘ Counterexample ’. Eqn : ‘ equation ’. RhS: ‘ RightHand Side ’ of an eqn or inequality. LhS: lefthand side ’. Sqrt : square-root ’, e.g, “the sqrt of 16 is 4”. Cts : ‘ continuous ’ and cty : ‘ continuity ’. Ptn : ‘ partition ’, but pt : ‘ point ’, as in “a fixed-pt of a map”. difference [ A r B ] [ B r A ]. Furthermore A u B , Sets A & B have at least one point in common; they intersect. A u B , The sets have no common point; dis- joint. The symbol “ A u B ” both asserts intersection and represents the set A B . For a collection C = { E j } j of sets in Ω, let the disjoint union F j E j or F ( C ) represent the union S j E j and also assert that the sets are pairwise disjoint.
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