  We report high resolution observations of the $^{12}$CO$(1\rightarrow0)$ and
$^{13}$CO$(1\rightarrow0)$ molecular lines in the Carina Nebula and the Gum 31
region obtained with the 22-m Mopra telescope as part of the The Mopra Southern
Galactic Plane CO Survey. We cover 8 deg$^2$ from $l = 285^{\circ}$ to
290$^{\circ}$, and from $b = -1.5^{\circ}$ to +0.5$^{\circ}$. The molecular gas
column density distributions from both tracers have a similar range of values.
By fitting a grey-body function to the observed infrared spectral energy
distribution from Herschel maps, we derive gas column densities and dust
temperatures. The gas column density has values in the range from $6.3\times\
10^{20}$ to $1.4\times 10^{23}$ cm$^{-2}$, while the dust temperature has
values in the range from 17 to 43 K. The gas column density derived from the
dust emission is approximately described by a log-normal function for a limited
range of column densities. A high-column density tail is clearly evident for
the gas column density distribution, which appears to be a common feature in
regions with active star formation. There are regional variations in the
fraction of the mass recovered by the CO emission lines with respect to the
total mass traced by the dust emission. These variations may be related to
changes in the radiation field strength, variation of the atomic to molecular
gas fraction across the observed region, differences in the CO molecule
abundance with respect to H$_{2}$, and evolutionary stage differences of the
molecular clouds that compose the Carina Nebula-Gum 31 complex.
