Current measurements of the Hubble constant H _ { 0 } on scale less than \sim 100 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 \sim 10 Mpc indicates that the matter density \Omega _ { 0 } is only \sim 0.2 – 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 } can not be determined from the m – z relation at moderate/high redshift unless the variation of the Hubble constant is 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 .