introduction-probability.pdf

R 1 h 2 r 2 but this implies h 1 h 2 and hence h 1 h

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r 1 = h 2 r 2 . But this implies h 1 h 2 and hence h 1 = h 2 and r 1 = r 2 . So it follows that (0 , 1] is the countable union of disjoint sets (0 , 1] = r (0 , 1] rational ( H r ) . If we assume that H ∈ B ((0 , 1]) then B ((0 , 1]) ⊆ L implies H r ∈ B ((0 , 1]) and λ ((0 , 1]) = λ r (0 , 1] rational ( H r ) = r (0 , 1] rational λ ( H r ) . By B ((0 , 1]) ⊆ L we have λ ( H r ) = λ ( H ) = a 0 for all rational numbers r (0 , 1]. Consequently, 1 = λ ((0 , 1]) = r (0 , 1] rational λ ( H r ) = a + a + . . . So, the right hand side can either be 0 (if a = 0) or (if a > 0). This leads to a contradiction, so H ∈ B ((0 , 1]).
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34 CHAPTER 1. PROBABILITY SPACES
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Chapter 2 Random variables Given a probability space (Ω , F , P ), in many stochastic models functions f : Ω R which describe certain random phenomena are considered and one is interested in the computation of expressions like P ( { ω Ω : f ( ω ) ( a, b ) } ) , where a < b. This leads us to the condition { ω Ω : f ( ω ) ( a, b ) } ∈ F and hence to random variables we will introduce now. 2.1 Random variables We start with the most simple random variables. Definition 2.1.1 [(measurable) step-function] Let (Ω , F ) be a mea- surable space. A function f : Ω R is called measurable step-function or step-function , provided that there are α 1 , ..., α n R and A 1 , ..., A n ∈ F such that f can be written as f ( ω ) = n i =1 α i 1I A i ( ω ) , where 1I A i ( ω ) := 1 : ω A i 0 : ω A i . Some particular examples for step-functions are 1I Ω = 1 , 1I = 0 , 1I A + 1I A c = 1 , 35
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36 CHAPTER 2. RANDOM VARIABLES 1I A B = 1I A 1I B , 1I A B = 1I A + 1I B - 1I A B . The definition above concerns only functions which take finitely many values, which will be too restrictive in future. So we wish to extend this definition. Definition 2.1.2 [random variables] Let (Ω , F ) be a measurable space. A map f : Ω R is called random variable provided that there is a sequence ( f n ) n =1 of measurable step-functions f n : Ω R such that f ( ω ) = lim n →∞ f n ( ω ) for all ω Ω . Does our definition give what we would like to have? Yes, as we see from Proposition 2.1.3 Let , F ) be a measurable space and let f : Ω R be a function. Then the following conditions are equivalent: (1) f is a random variable. (2) For all -∞ < a < b < one has that f - 1 (( a, b )) := { ω Ω : a < f ( ω ) < b } ∈ F . Proof . (1) = (2) Assume that f ( ω ) = lim n →∞ f n ( ω ) where f n : Ω R are measurable step-functions. For a measurable step- function one has that f - 1 n (( a, b )) ∈ F so that f - 1 (( a, b )) = ω Ω : a < lim n f n ( ω ) < b = m =1 N =1 n = N ω Ω : a + 1 m < f n ( ω ) < b - 1 m ∈ F . (2) = (1) First we observe that we also have that f - 1 ([ a, b )) = { ω Ω : a f ( ω ) < b } = m =1 ω Ω : a - 1 m < f ( ω ) < b ∈ F so that we can use the step-functions f n ( ω ) := 4 n - 1 k = - 4 n k 2 n 1I { k 2 n f< k +1 2 n } ( ω ) . Sometimes the following proposition is useful which is closely connected to Proposition 2.1.3.
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2.2. MEASURABLE MAPS 37 Proposition 2.1.4 Assume a measurable space , F ) and a sequence of random variables f n : Ω R such that f ( ω ) := lim n f n ( ω ) exists for all ω Ω . Then f : Ω R is a random variable.
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