5-3

Question 5.3:
Let \{f_k\} be a sequence of nonnegative measurable functions defined on E. If f_k \to f and f_k \leq f a.e. on E, show that \int_E f_k \to \int_E f. 
Solutions:
By Theorem 4.12, if \{f_k\} is a sequence of measurable functions and \lim_{k\to \infty} f_k exists a.e., it is measurable. Since we are given that \lim_{k\to \infty} f_k=f, we can say that f is indeed a nonnegative measurable function.
Case 1: If \int_E f is finite, then we can let \phi=f and apply the Lebesgue's Dominated Convergence Theorem for Nonnegative Functions (Theorem 5.19). Since f is a measurable function such that f_k \leq f a.e. for all k and \int_E f is finite, we can conclude that \int_E f_k \to \int_E f.
Case 2: If \int_E f=\infty, then the fact that f_k \to f implies that \int_E f_k \to \infty. Thus, we show that \int_E f_k \to \int_E f. 
This concludes the proof.