Prove Lebesgue Measure of f-g over [a,b]

[matt_747]

Assume f > g two measurable function on [a,b] and
E={(x,y) : g(x)≤y≤f(x), x ∈ [a,b]}
Show ∫ (f-g) dm = m×m E . that m is lebesgue measure.
(limits of integration is a,b)

help will be appreciated so much

 

[rochfor1]

The trick here is to use Tonelli's Theorem. Write m x m(E) as the integral of the characteristic function of E, then rewrite as an iterated integral first with respect to y, then with respect to x. From that point, try to write the characteristic function of E in a different way and the answer should fall out.