  We study the measure theory of a two-site quantum random walk. The truncated
decoherence functional defines a quantum measure $\mu_n$ on the space of
$n$-paths, and the $\mu_n$ in turn induce a quantum measure $\mu$ on the
cylinder sets within the space $\Omega$ of untruncated paths. Although $\mu$
cannot be extended to a continuous quantum measure on the full $\sigma$-algebra
generated by the cylinder sets, an important question is whether it can be
extended to sufficiently many physically relevant subsets of $\Omega$ in a
systematic way. We begin an investigation of this problem by showing that $\mu$
can be extended to a quantum measure on a "quadratic algebra" of subsets of
$\Omega$ that properly contains the cylinder sets. We also present a new
characterization of the quantum integral on the $n$-path space.
