I have been working on (from Folland Real Analysis p.52):
Exercise 15. If $\{f_n\}\subset L^+$, $f_n$ decreases pointwise to $f$, and $\int f_1 <\infty$, then $\int f = \lim \int f_n$.
I haven't been able to show that $\int f \geq \lim\int f_n$ (the other direction I think is just a simple application of Fatou, so I don't need help with that part).
What I have tried/considered:
This looks a lot like the Monotone Convergence theorem so I was trying to create an analogue of the proof of that (from p.50) in this case but I don't really know if that works here (I don't really know what the analogue of the sets $E_n:= \{x\in X\mid f_n\geq \alpha\phi(x)\}$ ($\alpha\in (0,1)$ fixed) from that proof would be in this case). I was also trying to just use simple functions and somehow take a supremum but I haven't been able to work it out that way either. Also it doesn't seem like (for the $\geq$ direction I need to prove yet) any of the preceding exercises or theorems are helpful here (other than Proposition 2.13 and possibly other than Proposition 2.20).
Notation:
In the above exercise 15. we are fixing a measurable space $(X,\mathcal{M})$ and we set $L^+$ to be the set of measurable functions $f:X\to [0,\infty]$ (and I think by this, Folland means $(\mathcal{M},\mathcal{B}_{\mathbb{R}})$-measurable).
Best Answer
Here's a proof without dominated convergence.
By manipulation on a set of measure zero, we may assume that $f_n, f : X \to [0, \infty )$. Moreover, the sequence $f_1 - f_n$ is an increasing sequence which converges pointwise to $f_1 - f$. By the monotone convergence theorem, it follows that:
$$\int (f_1 - f_n) \to \int (f_1 - f)$$
Thus, $$ \begin{align*} \int f &= \int f + \int (f_1 - f) - \int (f_1 - f) \\ &= \int f_1 - \lim_n\int f_1-f_n \\ &= \lim_{n \to \infty} \int f_n \end{align*}$$