Let $\{f_n\}$ be a sequence of nonnegative measurable functions on $E$ that converges pointwise on $E$ to $f$. Suppose $f_n \leq f$ on $E$ for each $n$. Show that $\lim\limits_{n \to \infty} \int_E f_n=\int_E f$
This can almost use the Monotone Convergence Theorem, but the sequence is nonnegative. Can I just choose a subsequence, $\{f_{n_k}\}$ of $\{f_n\}$ that is increasing or is there a way to construct an increasing sequence.?
Best Answer
Let $g_n = \inf_{m \geq n} f_m$. Then the $g_n$ are increasing to $f$ pointwise and $g_n \leq f_n \leq f$. Apply MCT.