  Current measurements of the Hubble constant $H_0$ on scale less than
$\sim100$ Mpc appear to be controversial, while the observations made at high
redshift seem to provide a relatively low value. On the other hand, the Hubble
expansion is driven by the matter content of the universe. The dynamical
analysis on scale of a few $\sim10$ Mpc indicates that the matter density
$\Omega_0$ is only $\sim0.2$--$0.3$, which is significantly smaller than
$\Omega_0=1$ predicted in the standard inflation model. This might support the
tendency of a decreasing Hubble constant towards distance. In this paper, we
discuss the influence of a possible variant Hubble constant on two fundamental
relations in astronomy: the magnitude-redshift ($m$--$z$) and the
number-magnitude relations. Using a distant type Ia supernova at $z=0.458$, we
show that the deceleration parameter $q_0$ or $\Omega_0$ cannot be determined
from the $m$--$z$ relation at moderate/high redshift unless the variation of
the Hubble constant is {\it a priori} measured. It is further demonstrated that
the number density of distant sources would be underestimated when their local
calibration is employed, which may partially account for the number excess of
the faint blue galaxies observed at moderate/high redshift.
