314 CHAPTER 10. THE ABSTRACT LEBESGUE INTEGRAL

23. Use Jensen’s inequality in Corollary 10.15.2 to show that if f is nonnegative andmeasurable, then for p > 1 show that whenever µ is a finite measure, then if f p ∈L1 (Ω) it follows that f ∈ L1 (Ω). Give an example to show that this is not necessarilytrue if µ (Ω) = ∞. Hint: For the second part, you might consider Ω = N, the σ

algebra the set of all subsets, and µ (S) equal to the number of elements in S. Maybef (n) = 1/n.