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Proof of 2 sets being equal.

  1. Feb 2, 2012 #1
    Hey there. I'm asked to prove:

    If X1,...Xn are random variables defined on a set Ω and B1,...,Bn C R1 then prove that
    (X1,...,Xn)^-1 (B1x...xBn) = X1^-1B1 n X2^-1B2 n ... n Xn^-1Bn

    so I think I can explain the proof, but just not write it out. This is my attempt
    if omega is on the left hand side:

    the inverse image of X1^-1B1 is defined as all w in Ω such that X1(w) E B1 and that would apply to all Xi, Bi.
    So the intersection of those events would be defined as all the w in Ω that
    (X1,...,Xn)(w) E (B1x...xBn) which is the left hand side.
     
  2. jcsd
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