$\lambda$ is given as any positive Borel measure on $R^{k}$ and $\lambda\left(E\right)=m\left(E\right)$ for all boxes $E$.
A box is defined in 2.19 of RCA (Please let me know if I should provide it here). The measure $m$ satisfies the properties of the Riesz representation theorem (Theorem 2.14).
Rudin concludes that $\lambda$ is regular using Theorem 2.18. This theorem (2.18) gives conditions for $\lambda$ to be regular which is that every open set is $\sigma$-compact (countable union of compact sets) and $\lambda\left(K\right)<\infty$, where $K$ is any compact set belong to the corresponding $\sigma$-algebra.
But I am not sure how Theorem 2.18 can be used as above (as it is done in Theorem 2.20) without knowing that $\lambda\left(K\right)<\infty$, where $K$ is any compact set belong to the corresponding $\sigma$-algebra.
Any pointers would be highly appreciated. I could be missing something simple, hence am happy to delete the question if this is too trivial.
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Best Answer
Any compact set $K$ is subset of some box $B$. So $\lambda (K) \leq \lambda (B)=m(B) <\infty$.
Any compact set is bounded. Hence it is contained in a box $𝑄\left(0,\delta\right)$ for large enough $\delta$. Hence, $\lambda\left(Q\left(0,\delta\right)\right) =m\left(Q\left(0,\delta\right)\right) = \delta^{k}<\infty$. Here, we are using property (a) of Theorem 2.20, which is that $m\left(W\right)=\text{vol}\left(W\right)$ for every $k$-cell $W$.