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Math Help - Fubini-Tonelli with complete measure space

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    Fubini-Tonelli with complete measure space

    So let (X,M,\mu), (Y,N,\nu) be complete \sigma-finite measure spaces. Then consider (X\times Y,L,\lambda), the completion of (X\times Y,M\times N,mu\times\nu).

    This is the basic set up. I have to show that if f is L-measurable and f=0 \lambda almost everywhere, then f_x,f^y are integrable and \int f_xd\nu=\int f^yd\mu=0 almost everywhere.

    Note: f_x(x,y)=f^y(x,y)=f(x,y) for fixed x,y.

    I was told that for this part that I need to use the fact that \mu,\nu are both complete but I don't see how.

    I am starting out by assuming that f=\chi_E (characteristic function). Then f_x=E_x right? I assume that it is at this point I need to make use of completeness some how.
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    Quote Originally Posted by putnam120 View Post
    So let (X,M,\mu), (Y,N,\nu) be complete \sigma-finite measure spaces. Then consider (X\times Y,L,\lambda), the completion of (X\times Y,M\times N,mu\times\nu).

    This is the basic set up. I have to show that if f is L-measurable and f=0 \lambda almost everywhere, then f_x,f^y are integrable and \int f_xd\nu=\int f^yd\mu=0 almost everywhere.

    Note: f_x(x,y)=f^y(x,y)=f(x,y) for fixed x,y.

    I was told that for this part that I need to use the fact that \mu,\nu are both complete but I don't see how.

    I am starting out by assuming that f=\chi_E (characteristic function). Then f_x=E_x right? I assume that it is at this point I need to make use of completeness some how.
    I don't see that completeness should be needed for this. According to Halmos, the definition of \lambda(E) is \lambda(E) = \textstyle\int\nu(E_x)\,d\mu(x) = \int\mu(E^y)\,d\nu(y) (of course, he has to show that those two integrals are equal). So the result for f = \chi_E follows straight from that definition. See 36 of Halmos's book, pp.145148. His results for product measures assume throughout that the measures on the component spaces are σ-finite, but not that they are complete.
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