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Homework Help: Proving functions in product space are measurable.

  1. Dec 2, 2012 #1
    1. The problem statement, all variables and given/known data
    I have a lot of questions that ask me to prove certain functions are measureable.

    For example I have to show that given f:X→ ℝ is M - measurable and g:Y→ ℝ is N - measurable
    implies that fg is M×N measurable.

    Another is prove that f = {1 when x=y, 0 else} is measurable on B[itex]_{[0,1]}[/itex]×P([0,1]) where B[itex]_{[0,1]}[/itex] is the borel sets on [0,1] with respect to lebesgue measure and the measure P([0,1]) is the counting measure (cardinality of a set in [0,1])

    2. Relevant equations
    Don't know any.

    3. The attempt at a solution

    I don't have a clue what to do because I don't know any definition of measurable functions in a product space. I know the case for single measure spaces, If E[itex]\in[/itex]N and f[itex]^{-1}[/itex](E) [itex]\in[/itex] M, then f is (M,N) - measurable.

    So for the first question all I know is that E[itex]\in[/itex]B[itex]_{ℝ}[/itex] and f[itex]^{-1}[/itex](E) [itex]\in[/itex] M, and F[itex]\in[/itex] B[itex]_{ℝ}[/itex] and g[itex]^{-1}[/itex](F) [itex]\in[/itex] N
  2. jcsd
  3. Dec 2, 2012 #2


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    Yes, but you are free to choose E=F here. What will the preimage of that be under (f, g)?
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