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Analysis + Fubini's Theorem

  1. Mar 1, 2006 #1
    I am having trouble with this problem involving Fubini's Theorem. I have done a question already similar to this ( i will post it as well ), but this question is a bit different, which is causing the problem.

    (question that i have completed)

    Let f be the function on Rdefined by
    [tex] f(x)=\left{\begin{array}{cc}1,&\mbox{ if } x is rational \\0, & \mbox{ if } x is irrational \end{array}\right [/tex]
    show that [tex] \int f(x)dx [/tex] does not exist for any a<b
    (a and b are the endpoints for the integral)

    consider inf sums and sup sums.

    Inf sum: I = [tex] \sum_\alpha (inf f) \Delta x [/tex]

    sup sum: S = [tex] \sum_\alpha (sup f) \Delta x [/tex]

    inf-sum [tex]\leq \int f \leq [/tex] sup-sum

    therefor by inspection
    inf f = 0 and sup f = 1
    I = [tex] \int f_{inf} = \int 0dx = 0 [/tex]
    S = [tex] \int f_{sup} = \int 1dx = 1 [/tex]

    so S-I cannot be made < [tex] \epsilon [/tex]
    ( now the problem i cant seem to figure out )

    Let f be the function on [tex] R^2 [/tex] defined by

    [tex] f(x)=\left{\begin{array}{cc}1,&\mbox{ if } x is rational/ and y =0, 1/2, or 1\\0, & \mbox otherwise\end{array}\right [/tex]

    and R the square R = {(x,y) [tex] are in R^2: 0 \leq x \leq 1, 0\leq y \leq 1} [/tex] determine if the integral exists.

    show that [tex] \int_R f(x)dxdy [/tex]
    that is the question... now it is similar to the one i have done but y is involved.. now im confused.. because does it make that much of a difference.
    i know i have to consider the inf and sup sums again. but what i dont know is the values of the inf and sup sums. Is it 0,1 again. any help is amazing and greatly appreciated.
    (in this problem instead of integrating once i would do it twice.. now would that give me 2 inf and 2 sup sums.. or somthing different?)

    also if there is any confusion ill try to clear it up??
    Last edited: Mar 1, 2006
  2. jcsd
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