  We present new determinations of the local sub-mm luminosity functions. We
find the local sub-mm luminosity density converging to 7.3+/-0.2 x 10^19
W/Hz/Mpc^3 /h_65 at 850um solving the ``sub-mm Olbers' Paradox.'' Using the
sub-mm colour temperature relations from the SCUBA Local Universe Galaxy
Survey, and the discovery of excess 450um excess emission in these galaxies, we
interpolate and extrapolate the IRAS detections to make predictions of the SEDs
of all 15411 PSC-z galaxies from 50-3000um. Despite the long extrapolations we
find excellent agreement with (a) the 90um luminosity function of Serjeant et
al. (2001), (b) the 850um luminosity function of Dunne et al. (2000), (c) the
mm-wave photometry of Andreani & Franceschini (1996); (d) the asymptotic
differential and integral source count predictions at 50-3000um by
Rowan-Robinson (2001). Remarkably, the local luminosity density and the
extragalactic background light together strongly constrain the cosmic star
formation history for a wide class of evolutionary assumptions. We find that
the extragalactic background light, the 850um 8mJy source counts, and the
Omega_* constraints all independently point to a decline in the comoving star
formation rate at z>1.
