  The Press-Schechter (PS) formalism for mass functions of the gravitationally
collapsed objects is reanalyzed. It has been suggested by many authors that the
PS mass function agrees well with the mass function given by $N$-body
simulations, while many too simple assumptions are contained in the formalism.
In order to understand why the PS formalism works well we consider the
following three effects on the mass function: the filtering effect of the
density fluctuation field, the peak ansatz in which objects collapse around
density maxima, and the spatial correlation of density contrasts within a
region of a collapsing halo because objects have non-zero finite volume.
According to the method given by Yano, Nagashima & Gouda (YNG) who used an
integral formula proposed by Jedamzik, we resolve the missing link in the PS
formalism, taking into account the three effects. While YNG showed that the
effect of the spatial correlation alters the original PS mass function, in this
paper, we show that the filtering effect almost cancels out the effect of the
spatial correlation and that the original PS is almost recovered by combining
these two effects, particularly in the case of the top-hat filter. On the other
hand, as for the peak ansatz, the resultant mass functions are changed
dramatically and not in agreement especially at low-mass tails in the cases of
the Gaussian and the sharp $k$-space filters. We show that these properties of
the mass function can be interpreted in terms of the kernel probability
$P(M_{1}| M_{2})$ in the integral formula qualitatively.
