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I am having difficulty understanding a step in the proof of Riesz Representation Theorem, in Rudin's 'Real and Complex Analysis' (P.40, Theorem 2.14):

Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_c(X)$ (the set of all continuous functions on X with compact support)

Rudin defines $$\mu(V)=sup\{\Lambda f:f\prec V\}$$ for every open set $V$ in $X$, where $f\prec V$ means $f$ is continuous and has compact support and $0\le f\le \chi_V$

He goes on to assert that $\mu(E)=inf\{ \mu(V): E\subseteq V, V \:open\}$ for every open set $E$ in $X$

But I have difficulty understanding why the above equality holds. I can prove $\mu(E)\leq inf\{ \mu(V): E\subseteq V, V \:open\}$ immediately from the definition of $\mu$ but the other side of the inequality is giving me trouble.

Any help in elaborating Rudin's statement is very much appreciated.

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$E$ is open and $E\subseteq E$, so $\mu(E)\in\{\mu(V):E\subseteq V, V \mathrm{open}\}$. Hence, $\mu(E)\geq \inf\{\mu(V):E\subseteq V, V \mathrm{open}\}$.

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  • $\begingroup$ Oh of course, how could I have overlooked that! Thanks! $\endgroup$ Aug 12, 2015 at 10:17
  • $\begingroup$ @Tsang The problem with the big Rudin book at the first reading is that some proof (like this one) are long and very difficult, and then we easily miss something ; so prepare to miss some easier points in the future! $\endgroup$
    – Nicolas
    Aug 12, 2015 at 10:19
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    $\begingroup$ @Nicolas -- AGREED! $\endgroup$
    – Procore
    Nov 2, 2017 at 22:32

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