The extended ROSAT-ESO Flux-Limited X-ray Galaxy Cluster Survey (REFLEX
  II) VII The Mass Function of Galaxy Clusters
  The mass function of galaxy clusters is a sensitive tracer of the
gravitational evolution of the cosmic large-scale structure and serves as an
important census of the fraction of matter bound in large structures. We obtain
the mass function by fitting the observed cluster X-ray luminosity distribution
from the REFLEX galaxy cluster survey to models of cosmological structure
formation. We marginalise over uncertainties in the cosmological parameters as
well as those of the relevant galaxy cluster scaling relations. The mass
function is determined with an uncertainty less than 10% in the mass range 3 x
10^12 to 5 x 10^14 M$_\odot$. For the cumulative mass function we find a slope
at the low mass end consistent with a value of -1, while the mass rich end
cut-off is milder than a Schechter function with an exponential term exp($-
M^\delta$) with $\delta$ smaller than 1. Changing the Hubble parameter in the
range $H_0 = 67 - 73 km s^-1 Mpc^{-1}$ or allowing the total neutrino mass to
have a value between 0 - 0.4 eV causes variations less than the uncertainties.
We estimate the fraction of mass locked up in galaxy clusters: about 4.4% of
the matter in the Universe is bound in clusters (inside $r_200$) with a mass
larger than 10^14 M$_\odot$ and 14% to clusters and groups with a mass larger
than 10^13 M$_\odot$ at the present Universe. We also discuss the evolution of
the galaxy cluster population with redshift. Our results imply that there is
hardly any clusters with a mass > 10^15 M$_\odot$ above a redshift of z = 1.
