I'm currently working on understanding The Reisz Representation Function in Papa Rudin. I'm wondering why we choose as our seed of a measure to be $$\hspace{-2in} (1) \hspace{2in} \mu(V) = \sup\{ \Lambda f : f \prec V \}. $$
I think this answer here does a great job of giving an intuition why. Although I'm stuck on one part of it. He says that for an open set A we can find continuous functions $f_n$ that approach $\chi_A$. Then $\lim_{n \rightarrow \infty}(\Lambda f_n)$ would be a good starting point for the value of $\mu(V)$. We then extend the measure to be defined for more sets to make a $\sigma$-algebra. My question is about $(1)$. How can we show that $f$ such that $f \prec V$ is arbitrary close to $\chi_V$? In other words, how do we know there is a sequence of functions $f_n$ such that $f_n \rightarrow \chi_V$ as $n \rightarrow \infty$. Then that would motivate the definition of $(1).$