text: {[ promptMessage ]}
  
  Bookmark it
  
  {[ promptMessage ]}
  
  solutions-1
  
  # solutions-1 - Solution to Homework#1 36-754 27 January 2006...
  
  This preview shows pages 1–2. Sign up to view the full content.
  
  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 .
  
  This preview has intentionally blurred sections. Sign up to view the full version.
  
  View Full Document
  This is the end of the preview. Sign up to access the rest of the document.
  
  {[ snackBarMessage ]}
  
  ### Page1 / 2
  
  solutions-1 - Solution to Homework#1 36-754 27 January 2006...
  
  This preview shows document pages 1 - 2. Sign up to view the full document.
  
  View Full Document
  Ask a homework question - tutors are online