I've been analyzing the proof of Riesz Representation Theorem as presented in Rudin's book for Real and Complex Analysis as Theorem 6.19 and found some steps confusing.
First I'd like to know why we can assume that $||\Phi||=1$. It seems to be vital step for proof but it's not fully clear for me why we do so. My first thought was that we can consider functions divided by measure of $X$, since they still belong to $C_0(X)$ and furthermore complex measure can take only finite values. Thus we have $f:=g/\mu(X)$ and
$$|\Phi f| \leqslant \int_X |f| d\mu= \frac{1}{|\mu(X)|} \int_X |g| d\mu \leqslant 1$$ for some $g\in C_0(X)$ with sup norm $||g||\leqslant 1$
Is this valid argument?
Second thing is why we can write $$\lambda(X)=\sup\{\Lambda f: 0\leqslant f\leqslant 1\; f\in C_c(X)\}?$$ It's obvious that $\sup\{\Lambda f: 0\leqslant f\leqslant 1\; f\in C_c(X)\} \leqslant \lambda(X)$, but how can I prove opposite inequality?
These may seem really trivial questions, yet I cannot figure them out.
Thanks in advance, K.
First question: you want to prove that $\Phi$ is represented by an integral. If you can do it for $\Phi/\|\Phi\|$, then you multiply the measure you obtain by $\|\Phi\|$ and you get a measure for $\Phi$.
For your second question, there is nothing to prove, as it is just the definition given in Theorem 2.14. With a different notation, though: in Theorem 2.14 he writes $f\prec X$, and on page 131 he states explicitly what that means.