  We measure the weak gravitational lensing shear power spectra and their
cross-power in two photometric redshift bins from the Canada-France-Hawaii
Telescope Lensing Survey (CFHTLenS). The measurements are performed directly in
multipole space in terms of adjustable band powers. For the extraction of the
band powers from the data we have implemented and extended a quadratic
estimator, a maximum likelihood method that allows us to readily take into
account irregular survey geometries, masks, and varying sampling densities. We
find the 68 per cent credible intervals in the $\sigma_8$-$\Omega_{\rm
m}$-plane to be marginally consistent with results from $Planck$ for a simple
five parameter $\Lambda$CDM model. For the projected parameter $S_8 \equiv
\sigma_8(\Omega_{\rm m}/0.3)^{0.5}$ we obtain a best-fitting value of $S_8 =
0.768_{-0.039}^{+0.045}$. This constraint is consistent with results from other
CFHTLenS studies as well as the Dark Energy Survey. Our most conservative
model, including modifications to the power spectrum due to baryon feedback and
marginalization over photometric redshift errors, yields an upper limit on the
total mass of three degenerate massive neutrinos of $\Sigma m_\nu < 4.53 \,
{\rm eV}$ at 95 per cent credibility, while a Bayesian model comparison does
not favour any model extension beyond a simple five parameter $\Lambda$CDM
model. Combining the shear likelihood with $Planck$ breaks the
$\sigma_8$-$\Omega_{\rm m}$-degeneracy and yields $\sigma_8=0.818 \pm 0.013$
and $\Omega_{\rm m} = 0.300 \pm 0.011$ which is fully consistent with results
from $Planck$ alone.
