A monotone convergence theorem for nets in Reed and Simon, Vol. I Reed and Simon state the following monotone convergence theorem for nets (Theorem IV.15, Vol. I).

Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$. Let $\{f_\alpha\}_{\alpha\in I}$ be an increasing net of continuous functions. Then, $f=\lim_\alpha f_\alpha \in L^1(X,d\mu)$ if and only if $\sup_\alpha \| f_\alpha \|_1 < \infty$ and in that case $\lim_\alpha \|f-f_\alpha\|_1 = 0 $.

Unfortunately, they do not give a proof. Can anyone provide one, or give a reference to one? In the notes to the chapter containing the theorem, Reed and Simon cite Bourbaki's Integration, but I've found that text difficult to navigate. I do not have access to the other text they cite (Nachbin).
This is not quite your classical monotone convergence theorem, as it requires each $f_\alpha$ to be continuous. This is necessary in order to rule out common counter-examples (see, for instance, this question or these answers) to the statement of the dominated convergence theorem for nets instead of sequences.
I could try adapting a usual proof for the MCT. However, I don't quite see the role that continuity of the $f_\alpha$'s play in the result, so I'm not sure how to make use of it.
Also, I'm not actually certain that the hypotheses of the theorem rule out the example here. I suspect the monotonicity requirement would get in the way of choosing the net of functions required in the example, but I don't have a proof of that either.
Reference
Reed, Michael; Simon, Barry, Methods of modern mathematical physics. I: Functional analysis. Rev. and enl. ed, New York etc.: Academic Press, A Subsidiary of Harcourt Brace Jovanovich, Publishers, XV, 400 p. $ 24.00 (1980). ZBL0459.46001.
 A: Here is a proof with the added assumption that for each $\alpha \in I$, $f_\alpha \geq 0$ (but without the assumption that $X$ is compact). In this case, all of the integrals involved are automatically well-defined as elements of $[0,\infty]$ and the result boils down to showing that $$\int f d\mu = \sup_{\alpha \in I} \int f_\alpha d\mu$$
since it is easy to check that the right hand side is the same as $\lim_\alpha \int f_\alpha d \mu$. In fact, it is clear that $\int f d\mu \geq \sup_\alpha \int f_\alpha d\mu$ so we only have the other inequality to show.
If $U_j^n = f^{-1}((j2^{-n}, \infty))$ then
$$\varphi_n = 2^{-n} \sum_{j=1}^{2^{2n}} \chi_{U_j^n}$$ 
is a sequence of simple functions increasing to $f$ (this is one of the standard constructions of such a sequence). The key point is that since $f = \sup_{\alpha} f_\alpha$ and the $f_\alpha$ are continuous, $f$ is lower semicontinuous and so each $U_j^n$ is open. This will allow us to leverage regularity of $\mu$.
Fix $\varepsilon > 0$. For $n$ sufficiently large, by the M.C.T. for sequences, we have that $\int \varphi_n d \mu > \int f d\mu - \varepsilon$. Fix also an $n$ large enough such that this inequality holds. 
By regularity of the measure $\mu$, we can find compact sets $K_j \subset U_j^n$ such that if $\Phi = 2^{-n} \sum_{j=1}^{2^{2n}} \chi_{K_j}$ then $\int \Phi d\mu > \int f d\mu - \varepsilon$ also. Then, since $f > \varphi_n \geq \Phi$, for each $x \in X$ there is an $\alpha_x$ such that $f_{\alpha_x}(x) > \Phi(x)$. The point of this construction is that since $X$ is Hausdorff, each $K_j$ is closed so that $\Phi$ is upper-semicontinuous. Hence $f_{\alpha_x} - \Phi$ is lower semicontinuous and so 
$$O_x = \{y: f_{\alpha_x}(y) - \Phi(y) > 0\}$$
is open. Now $\{O_x: x \in X\}$ is an open cover for the compact set $\bigcup_{j=1}^{2^{2n}} K_j$ and hence there are $x_1, \dots, x_m$ such that $\{O_{x_i}: i = 1, \dots, m\}$ is also an open cover.
We can pick an $\alpha_0 \in I$ such that $\alpha_0 \geq \alpha_{x_i}$ for each $i$. This guarantees that $f_{\alpha_0} - \Phi \geq 0$ everywhere. Hence
$$\sup_\alpha \int f_\alpha d \mu \geq \int f_{\alpha_0} d \mu \geq \int \Phi d \mu \geq \int f d \mu - \varepsilon.$$
$\varepsilon > 0$ was arbitrary so the result follows.

In the comments, I gave an example that shows that with only the assumptions given in the question the result 
"$f \in L^1(\mu)$ if and only if $\sup_\alpha \|f_\alpha\|_1 < \infty$" is incorrect.
However, in the above, I don't use the assumption that $X$ is compact. With this assumption, one can show that with the assumptions given in the question the result
"$f \in L^1(X, d\mu)$ if and only if for every $\alpha_0 \in I$, $\sup_{\alpha \geq \alpha_0} \|f_\alpha\|_1 < \infty$" is true.
The key point is that this tells us that for each $\alpha_0$, $\min_{x \in X} f_{\alpha_0}(x) > - \infty$ and that $\mu(X) < \infty$ by regularity of $X$. The result I give then follows by applying the result already proved in this question with functions of the form $f_\alpha - \min_{x \in X} f_{\alpha_0}(x)$.
