The spatial-temporal averaging procedure is considered with a nonhomog
eneous distribution of elementary domains in the spatial-temporal spac
e and the probabilistic interpretation of the ST-averaging is also giv
en. Several averaging theorems and corollaries about the averages of s
patial and temporal derivatives are presented and rigorously proved wh
ich allow elementary domain to vary in space and time. The macroscopic
transport equation in the most general condition and the simplified m
acroscopic equation under the special form of distributions are develo
ped which may be reduced to the classical macroscopic transport equati
on as the spatial-temporal average degenerates into the volume average
.