I was studying the details of integration on null hypersurfaces through A Relativist's Toolkit by Eric Poisson.
If we consider a null hypersurface in 4-dimensional spacetime, then the surface element, according to Poisson, is given by$$ \mathrm{d}\Sigma_\alpha=-k_\alpha \sqrt{\sigma} \, \mathrm{d}^2\theta \, \mathrm{d}\lambda\,, $$where $k$ is a future pointing vector tangent to the surface along the null generators, $\theta$ are the coordinates transverse to the null generators and $\lambda$ parametrizes the null generators.
If we are in two dimensional spacetime, the null hypersurfaces are simply null geodesics. In this case we don't have transverse directions to the generators and hence I expect that integrating any smooth function along a null geodesic should yield zero. Do you agree with this statement?