I'm trying to prove the Monotone Convergence Theorem for decreasing sequences, namely if
Let $(X,\mathcal{M},\mu)$ be a measure space and suppose $\{f_n\}$ are non-negative measurable functions decreasing pointwise to $f$. Suppose also that $f_1 \in \mathscr{L}(\mu)$. Then $$\int_X f~d\mu = \lim_{n\to\infty}\int_X f_n~d\mu.$$
Why does this statement not follow from LDCT with $f_n$ being dominated by $f_1$?
I'm also aware of the solutions with $g_n=f_1-f_n$, but the question asks to prove it using Fatou's lemma
Best Answer
As you say in the comments, the problem is posed at the end of the first chapter of Rudin's Real and Complex Analysis, so it is not clear whether Rudin intends the student to use the Monotone Convergence Theorem, Fatou's Lemma or the Dominated Convergence Theorem.
You are right that the proof is a trivial application of DCT. However, it is certainly an instructive exercise to prove it using just MCT or Fatou's Lemma.