Necessity of absolute value in The Fubini–Tonelli theorem? In The Fubini–Tonelli theorem(reference in wiki):
What's the point of taking absolute value in $\int_{X\times Y}|f(x,y)|\,d(x,y)$? Isn't $f$ integrible in $X\times Y$ automatically implies $|f|$ integrable in $X\times Y$? So we only need $f$ integrable to apply  Fubini–Tonelli theorem?

No .The theorem holds for all non-negative measurable functions; you compute the repeated iterated integral of the absolute value of of f and if it comes out finite you CONCLUDE the $f$ is integrable so you may then apply Fubini to $f$ to calculate the integral of $f$ by iterated integrals -very useful result   
