I've been annotating the steps in Riesz Rep. Theorem. So far, I have almost all of them. I just have six questions, some are short questions. So these questions are for anyone who has Rudin Real and Complex analysis on hand.
(1) My first question starts on page $44$ for step IV. We start with two disjoint compact sets $K_1, K_2$. I'm not really clear on what $f(x) = 0$ on $K_2$. My reasoning is that we can an open set $V$ such that it contains $K_1$ and $V \cap K_2 = \emptyset$. So using Urysohn's lemma, we get $K_1 \prec f \prec V$ for some $f \in C_c(X)$. The support of $f$ lies in $V$, and since $K_2 \cap V = \emptyset$, $f(x) = 0$ for all $x \in K_2$. Are we guaranteed that we can find such open set $V$?
(2) My next question is on the same step but at equation $(13)$. So far, it says $(9)$ shows that $(13)$ holds. I don't see how this is obvious. For me, i started with step I, used that $\mu(E) \leq \sum_{1}^{\infty}\mu(E_i)$. I just use the fact that there must be infinite number of zero sets or else $\mu(E) < \infty$ is violated. Also there must be finite number of non-zero sets, so is that why we get $(13)$?
(3) On page 46 Step $X$. Rudin says, "Clearly, it is enough to prove this for real $f$". I don't have a complex analysis background but is he insinuating that the case for $f$ being complex is almost the same?
(4) On the same page, between equation $(18)$ and $(19)$, Rudin mentions that the sets $E_i$ are therefore disjoint Borel sets whose union is $K$. I understand this paragraph except the part that $E_i$ are Borel sets. I'm thinking that I have to verify that $E_i$ are either closed or open. We have that $f$ is continuous so the pre-image of an open set is open. But I am having trouble getting to verify that fact.
(5) In the same paragraph, Rudin says "There are open sets $V_i \supset E_i$ such that $\mu(V_i) < \mu(E_i) + \frac{\epsilon}{n}$" equation $(19)$ and such that $f(x) < y_i + \epsilon$ for all $x \in V_i$. Using equation $(2)$, I understand that $\mu(E_i) + \epsilon > \mu(V)$ for some open set $V$ such that $V \supset E$. Why does Rudin define equation $(19)$ as that way (the epsilon term divided by n) and was wondering why $f(x) < y_i + \epsilon$ for $x \in V_i$ <--- for this part, I can get a $V_i$ satisfying this inequality but dont see how it also satisfy $(19)$
(6) Same page, at the bottom, Rudin says that Step II shows that $\mu(K) \leq \Lambda(\sum h_i) = \sum \Lambda h_i$. This isn't obvious at all. I looked at $\mu(K) = \inf\{\Lambda f \mid K \prec f\}$ but we have that $h_i \prec V_i$. I tried showing that $K \prec \sum h_i$. But what seems to be the trouble is verifying that $0 \leq \sum h_i \leq 1$ for all $x \in X$
Thanks a bunch!