I just started looking through the proof of the Riesz Representation Theorem (in Rudin's Real and Complex Analysis), and I am still very confused about several things. I'll just write the statement of the theorem as given in the book:
Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_c(X).$ Then there exists a $\sigma$-algebra $M$ in $X$ which contains all Borel sets in $X$, and there exists a unique positive measure $\mu$ on $M$ which represents $\Lambda$ in the sense that,
(a) $\Lambda f = \int_Xf \ d\mu$ for every $x\in X$
(b) $\mu(K)<\infty$ for every compact set $K\subset X$
(c) For every $E\in M$, we have $\mu(E)=\inf\{\mu(V):E\subset V, V \text{ open}\}$
(d) The relation $\mu(E) = \sup\{\mu(K): K\subset E, K \text{ compact}\}$ holds for every open set $E$, and for every $E\in M$ with $\mu(E)<\infty$.
(e) If $E\in M$, $A\subset E$, and $\mu(E)=0$, then $A\in M$. "
The proof of this statement spans 7 pages, and while I was able to follow the logic step-by-step, I feel like I have left with very little intuition regarding the theorem as a whole. I am currently trying to solve the problem:
Show that
$ \Lambda f = \int_{-\infty}^\infty f(x) |x| dx, \ \ \ f\in C_c(\mathbb{R}) $
is a positive linear functional on $C_c(\mathbb{R})$. Denoting by $\mu$ the measure representing $\Lambda$, compute $\mu([0,\pi])$.
So I suppose my specific question is how do the restrictions on the measure given in $(b)-(d)$ allow us to explicitly calculate the value of $\mu$ for $[0,\pi]$ for the $\Lambda$ above? More generally, it would be nice to see another proof of the Riesz Representation Theorem, as Rudin's has not really sunk in yet. If you have any reference recommendations that give a more intuitive (or just different) proof, it'd be much appreciated.