I would a appreciate if someone could take the time to check if my solution to the following problem is correct:
From http://www.math.chalmers.se/~borell/MeasureTheory.pdf, page 64, ex.6.
Let (X,M,μ) be a positive measure space and suppose f and g are non-negative measurable functions such that ∫Afdμ=∫Agdμ,all A∈M.
(a). Prove that f=g a.e. [μ] if μ is σ-finite.
(b). Prove that the conclusion of Part (a) may fail if μ is not σ-finite.
Consider the set where f>g. Denote this set A. But A=∪n[An] where An={x:f(x)>g(x)+1/n},
In that case consider Cn={x:g(x)<n}.
Choose M s.t. μ(AM,XM,CM)>0, now integrate over this set instead to arrive at the desired contradiction. Hence μ(A)=0
The same applies to the set B={x:g(x)>f(x)},μ(B)=0.
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