introduction-probability.pdf

Finally we use that e f e f 2 e f 2 e f 2 e f 2 70

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Finally, we use that E ( f - E f ) 2 = E f 2 - ( E f ) 2 E f 2 .
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70 CHAPTER 3. INTEGRATION 3.7 Theorem of Radon-Nikodym Definition 3.7.1 (Signed measures) Let (Ω , F ) be a measurable space. ( i ) A map μ : F → R is called (finite) signed measure if and only if μ = αμ + - βμ - , where α, β 0 and μ + and μ - are probability measures on F . ( ii ) Assume that (Ω , F , P ) is a probability space and that μ is a signed measure on (Ω , F ). Then μ P ( μ is absolutely continuous with respect to P ) if and only if P ( A ) = 0 implies μ ( A ) = 0 . Example 3.7.2 Let (Ω , F , P ) be a probability space, L : Ω R be an integrable random variable, and μ ( A ) := A Ld P . Then μ is a signed measure and μ P . Proof . We let L + := max { L, 0 } and L - := max {- L, 0 } so that L = L + - L - . Assume that Ω L ± d P > 0 and define μ ± ( A ) = Ω 1I A L ± d P Ω L ± d P . Now we check that μ ± are probability measures. First we have that μ ± (Ω) = Ω 1I Ω L ± d P Ω L ± d P = 1 . Then assume A n ∈ F to be disjoint sets such that A = n =1 A n . Set α := Ω L + d P . Then μ + n =1 A n = 1 α Ω 1I n =1 A n L + d P = 1 α Ω n =1 1I A n ( ω ) L + d P = 1 α Ω lim N →∞ N n =1 1I A n L + d P = 1 α lim N →∞ Ω N n =1 1I A n L + d P
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3.8. MODES OF CONVERGENCE 71 = n =1 μ + ( A n ) where we have used Lebesgue ’s dominated convergence theorem. The same can be done for L - . Theorem 3.7.3 (Radon-Nikodym) Let , F , P ) be a probability space and μ a signed measure with μ P . Then there exists an integrable random variable L : Ω R such that μ ( A ) = A L ( ω ) d P ( ω ) , A ∈ F . (3.6) The random variable L is unique in the following sense. If L and L are random variables satisfying (3.6), then P ( L = L ) = 0 . The Radon-Nikodym theorem was proved by Radon 7 in 1913 in the case of R n . The extension to the general case was done by Nikodym 8 in 1930. Definition 3.7.4 L is called Radon-Nikodym derivative. We shall write L = d P . We should keep in mind the rule μ ( A ) = Ω 1I A = Ω 1I A Ld P , so that ’ = Ld P ’. 3.8 Modes of convergence First we introduce some basic types of convergence. Definition 3.8.1 [Types of convergence] Let (Ω , F , P ) be a probabil- ity space and f, f 1 , f 2 , ... : Ω R random variables. (1) The sequence ( f n ) n =1 converges almost surely (a.s.) or with prob- ability 1 to f ( f n f a.s. or f n f P -a.s.) if and only if P ( { ω : f n ( ω ) f ( ω ) as n → ∞} ) = 1 . 7 Johann Radon, 16/12/1887 (Tetschen, Bohemia; now Decin, Czech Republic) - 25/05/1956 (Vienna, Austria). 8 Otton Marcin Nikodym, 13/08/1887 (Zablotow, Galicia, Austria-Hungary; now Ukraine) - 04/05/1974 (Utica, USA).
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72 CHAPTER 3. INTEGRATION (2) The sequence ( f n ) n =1 converges in probability to f ( f n P f ) if and only if for all ε > 0 one has P ( { ω : | f n ( ω ) - f ( ω ) | > ε } ) 0 as n → ∞ . (3) If 0 < p < , then the sequence ( f n ) n =1 converges with respect to L p or in the L p -mean to f ( f n L p f ) if and only if E | f n - f | p 0 as n → ∞ . Note that { ω : f n ( ω ) f ( ω ) as n → ∞} and { ω : | f n ( ω ) - f ( ω ) | > ε } are measurable sets.
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