We consider a dynamical system with state space $M$, a smooth, compact subset
of some ${\Bbb R}^n$, and evolution given by $T_t$, $x_t = T_t x$, $x \in M$;
$T_t$ is invertible and the time $t$ may be discrete, $t \in {\Bbb Z}$, $T_t =
T^t$, or continuous, $t \in {\Bbb R}$. Here we show that starting with a
continuous positive initial probability density $\rho(x,0) > 0$, with respect
to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb
R}^n$, the expectation value of $\log \rho(x,t)$, with respect to any
stationary (i.e. time invariant) measure $\nu(dx)$, is linear in $t$, $\nu(\log
\rho(x,t)) = \nu(\log \rho(x,0)) + Kt$. $K$ depends only on $\nu$ and vanishes
when $\nu$ is absolutely continuous wrt $dx$.