Looking for proof of theorem on complex measurable functions

Question

In University I have been given the following result:

If $f:X\to\mathbb{C}$ is a measurable function in $L^1(X,\mathcal{E},\mu)$ with $\mu$ being finite, and there exists a closed set $S\subseteq\mathbb{C}$ for which the means of the functions over any positive-measure set belong to $S$, i.e. $\frac{1}{\mu(E)}\int\limits_{{}^E}f\mathrm{d}\mu\in S$ for every $E\in\mathcal{E}$ such that $\mu(E)>0$, then $f(x)\in S$ for $\mu$-almost every $x\in X$.

I haven't been given a proof of this theorem. I have at my disposal Rudin, Real and Complex Analysy, on which I have been given a wrong reference to what should be this result and is actually "Comments on Definition 1.2", Conway, A Course in Functional Analysis, and these notes of a course in Real and Functional Analysis, but am unable to find this result anywhere in any of these references. I was wondering:

1. Does this result carry a special name with it, like the Riesz representation theorem, the Carathéodory extension theorem etc? If so, what is it?
2. Where can I find a proof in any freely available book in any format or pdf of any kind (PS Google Books have given me problems, so I would disadvise you from linking to them as reference :) )? Alternately, can you answer with a proof?

I admit I haven't thought too much about this. I will in a while, though I'm not really sure I know where to start. If I find anything that seems to be getting anywhere, an edit will take place immediately.

Answer 1

This is exactly Theorem 1.40 in Rudin's "Real & complex analysis" (3rd ed.)

Answer 2

As noted by copper.hat's answer, the result is theorem 1.40 on Rudin's book. For anyone who should stumble upon this question, I place here a screenshot of the proof. I think this does not violate any kind of copyright, since the book can anyway be fully downloaded from the above provided link.