This is a theorem from Folland's Real Analysis, and I have difficulty understanding the proof.
My problem is the last two sentences. $H_n$ as defined is clearly compact, and I follow the inequality obtained, which comes from additivity of the measure. Also, taking the limit as $n\to \infty$, we clearly get the last inequality too, but the limit should also affect $H_n$ in the above inequality. However,$\lim_{n\to \infty} H_n=\bigcup_{-\infty}^{\infty}H_n$ need not be a compact set. So how is the result proven here? I would greatly appreciate some help to complete my understanding of this proof.
The point is not that $\bigcup\limits_{-\infty}^{\infty} K_j$ is compact, but that for each finite $n$, the union $H_n := \bigcup\limits_{-n}^n K_j$ is compact. What he proves can now be interpreted as $$\mu(E) \geq \sup\{\mu(K) : K\subset E, K \text{compact}\} \geq \sup\{\mu(H_n): n \in \Bbb{N}\} = \lim\limits_{n \to \infty} \mu(H_n) = \mu(E).$$