If f(x) > 0 for all x S and is -finite, then S f d

If f(x) > 0 for all x ∈ S and µ is σ-finite, then S f dµ = 0 implies

µ(S) = 0.

[Let Sn be the subset of S on which f(x) ≥ 1/n Then µ(S) ≤ µ(Sn) and

µ(Sn) ≤ n Sn f dµ ≤ n S f dµ = 0.]

Section 2.3