= Solution
A ring of sets is closed under finite unions and differences. A <premeasure> on it is a nonnegative set function with value zero on the empty set that is countably additive whenever a disjoint union remains in the ring. <Caratheodory extension theorem> extends a <premeasure> to a measure on the generated <sigma-algebra>; this extension is unique there when the <premeasure> is sigma-finite. It is obtained from the <outer measure>
$$
\mu^*(E)=\inf\left\{\sum_j\mu(A_j):E\subseteq\bigcup_jA_j,\ A_j\in\mathcal A\right\},
$$
whose measurable sets satisfy $\mu^*(E)=\mu^*(E\cap B)+\mu^*(E\setminus B)$ for every $E$. The resulting complete measurable <sigma-algebra> contains the <sigma-algebra> generated by the ring.
For the required continuity, let $B_n\downarrow\varnothing$ in the bounded-interval ring and fix $\epsilon>0$. From each component $(a,b]$ of $B_n$, remove a sufficiently short initial piece and retain the compact interval $[a+\delta,b]$. The union is a compact set $K_n\subset B_n$, and $B_n\setminus K_n$ is covered by a ring set $C_n$ of total length less than $\epsilon2^{-n}$; one can take unions of $(a,a+\delta]$. We do not incorrectly assume the compact sets themselves belong to the half-open interval ring.
Since $\bigcap_nK_n=\varnothing$ and $K_1$ is compact, the finite intersection property supplies $N$ with $\bigcap_{n=1}^NK_n=\varnothing$. For $x\in B_N$, membership in all earlier $B_n$ means some failure of membership in $K_n$ puts $x$ in $C_n$. Thus
$$
B_N\subseteq\bigcup_{n=1}^NC_n,\qquad
\mu(B_N)\leq\sum_{n=1}^N\mu(C_n)<\epsilon.
$$
Finite additivity gives the finite subadditivity used here. Monotonicity now proves \b[$\mu(B_n)\to0$].
If a disjoint union $A=\bigcup_jA_j$ lies in the ring, its remainders $A\setminus\bigcup_{j\leq n}A_j$ decrease to the empty set. The proved continuity and finite additivity imply countable additivity, so the length function is a <premeasure>. The ring generates the Borel <sigma-algebra> and covers the line by bounded intervals of finite measure. Its unique sigma-finite extension is Borel length measure; its completion is <Lebesgue measure>.
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