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Fubini's Theorem

  1. Oct 23, 2012 #1
    Does anybody know if the following is true?

    Let $p>1$. If $f=f(x,y)$ is such that $f\in L^p([a,b]\times [a,b])$, then $f_y(x)\in L^p([a,b])$ for almost all $y\in[a,b]$ and $f_x(y)\in L^p([a,b])$ for almost all $x\in[a,b]$.

    Is this a consequence of Fubini's theorem?
  2. jcsd
  3. Oct 23, 2012 #2
    Unfortunately, $ don't work. You need to enclose your code with [/itex] and [itex]. The one with the slash goes at the end but I had to put them in reverse order for it to show up (or it would have just treated "and" as my code.
  4. Oct 23, 2012 #3
    The answer to your question is yes. g is Lp if and only if |g|^p is L1. So apply Fubini to the function [itex]|f(x,y)|^p[/itex], which is L^1.
  5. Oct 23, 2012 #4
    Thanks a lot for help!!
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