# ans4-1 - ECON 831 Solution Homework #4 1. Bierens, 11...

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ECON 831 Solution Homework #4 1. Bierens, 11 Consider F an algebra. (i) First, we show that F σ -algebra ⇒ F monotone class. By de nition of a σ -algebra, we know that for any sequence of sets in F , their in nite union is also an element of F , and their in nite intersection is also an element of F . This is true in particular for increasing and decreasing sequences. Hence the two properties characterizing a monotone class are satis ed, and we conclude that F is a monotone class. (ii) Second, we show that F monotone class ⇒ F σ -algebra. By assumption F is an algebra, and we need to show that it is also a σ -algebra. That is, we need to show that for any sequence of sets in F , their in nite union is still an element of F . Consider then an arbitrary sequence of sets in F , say A n ∈ F . We now construct a new sequence, C n , s.t. C i = S i k =1 A k : so we have C 1 = A 1 , C 2 = A 1 A 2 , ... By de nition, the sequence C n is increasing, that is C n C n +1 , C n ∈ F , and also S i k =1 A k = S i k =1 C k . By assumption we know that F is a monotone class, so we can use one of its property to deduce that S k =1 C k ∈ F . And since S k =1 A k = S k =1 C k , we have S k =1 A k ∈ F . We conclude that F is a σ -algebra. 2. Bierens, 12 Consider F a λ -system. We show that F π -system ⇒ F σ -algebra. We need to check the three properties that de ne a σ -algebra. First, let me recall the three properties satisfy by a πλ -system: (1) A ∈ F ⇒ A c ∈ F

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## ans4-1 - ECON 831 Solution Homework #4 1. Bierens, 11...

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