  A discrete quantum process is defined as a sequence of local states $\rho_t$,
$t=0,1,2,...$, satisfying certain conditions on an $L_2$ Hilbert space $H$. If
$\rho =\lim\rho_t$ exists, then $\rho$ is called a global state for the system.
In important cases, the global state does not exist and we must then work with
the local states. In a natural way, the local states generate a sequence of
quantum measures which in turn define a single quantum measure $\mu$ on the
algebra of cylinder sets $\cscript$. We consider the problem of extending $\mu$
to other physically relevant sets in a systematic way. To this end we show that
$\mu$ can be properly extended to a quantum measure $\mutilde$ on a "quadratic
algebra" containing $\cscript$. We also show that a random variable $f$ can be
"quantized" to form a self-adjoint operator $\fhat$ on $H$. We then employ
$\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are
given
