I have a question concerning the affirmation : "There are compact sets $K_n$ and open sets $V_n$ such that $K_n \subset T_n \subset V_n \subset V$ and $µ(V_n-K_n)<2^{-n} \epsilon$."How do we know this is true ? I understand that according to theorem 2.14 (Riesz representation theorem) it is true for each set $E$ provided that $E$ is measurable and its measure is finite.So my question could be rephrased as follows : How do we know that each $T_n$ is measurable ?
I'm sorry I'm not yet allowed to insert images : but Rudin's RCA is very well-known and you can get it on the internet. Here it is :