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# solutions-1 - Solution to Homework#1 36-754 27 January 2006...

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Solution to Homework #1, 36-754 27 January 2006 Exercise 1.1 (The product σ -field answers count- able questions) Let D = S X S , where the union ranges over all countable sub- sets S of the index set T . For any event D ∈ D , whether or not a sample path x D depends on the value of x t at only a countable number of indices t . (a) Show that D is a σ -field. (b) Show that if A ∈ X T , then A ∈ X S for some countable subset S of T . Cf. the proof of Theorem 29 in the notes. (a): We must show that (i) Ξ T ∈ D , (ii) A ∈ D ⇒ Ξ T \ A ∈ D and (iii) A n ∈ D ⇒ n A n ∈ D for any countable collection of sets A n . (i): Pick S = { t } for any t T , and take the base set to be Ξ, i.e, the base set is x Ξ T : x t Ξ t . Clearly, this set is Ξ T . (ii): Fix S . Then for any A ∈ X S , Ξ T \ A = Ξ T \ A × t T \ S Ξ t = ( Ξ S \ A ) × t T \ S Ξ t which is in X S . (iii): Take any countable collection of sets A n ∈ D . For each such set, there is a corresponding finite set of indices, S n , for which A n ∈ X S n . Let S = n S n .

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The text appears to be a solution to a homework problem in measure theory or probability, specifically dealing with σ-fields and product spaces.
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the text does not contain a mathematical question but rather provides a solution to a homework problem. it explains the solution to a problem about a σ-field d, which is a collection of subsets of a product space. the solution demonstrates that d is a σ-field by verifying the three properties of a σ-field: it contains the entire space, it is closed under complementation, and it is closed under countable unions.
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# solutions-1 - Solution to Homework#1 36-754 27 January 2006...

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Solution to Homework #1, 36-754 27 January 2006 Exercise 1.1 (The product σ -field answers count- able questions) Let D = S X S , where the union ranges over all countable sub- sets S of the index set T . For any event D ∈ D , whether or not a sample path x D depends on the value of x t at only a countable number of indices t . (a) Show that D is a σ -field. (b) Show that if A ∈ X T , then A ∈ X S for some countable subset S of T . Cf. the proof of Theorem 29 in the notes. (a): We must show that (i) Ξ T ∈ D , (ii) A ∈ D ⇒ Ξ T \ A ∈ D and (iii) A n ∈ D ⇒ n A n ∈ D for any countable collection of sets A n . (i): Pick S = { t } for any t T , and take the base set to be Ξ, i.e, the base set is x Ξ T : x t Ξ t . Clearly, this set is Ξ T . (ii): Fix S . Then for any A ∈ X S , Ξ T \ A = Ξ T \ A × t T \ S Ξ t = ( Ξ S \ A ) × t T \ S Ξ t which is in X S . (iii): Take any countable collection of sets A n ∈ D . For each such set, there is a corresponding finite set of indices, S n , for which A n ∈ X S n . Let S = n S n .

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The text appears to be a solution to a homework problem in measure theory or probability, specifically dealing with σ-fields and product spaces.

TYPE: NO_QUESTION

The text does not contain a mathematical question but rather provides a solution to a homework problem. It explains the solution to a problem about a σ-field D, which is a collection of subsets of a product space. The solution demonstrates that D is a σ-field by verifying the three properties of a σ-field: it contains the entire space, it is closed under complementation, and it is closed under countable unions.