On a Riemannian manifold one can define the volume over the totality of the manifold $M$
$$ V=\int_M F(x,y,z,t)\sqrt{|g|}\mathrm dx\wedge\mathrm dy\wedge\mathrm{d}z\wedge \mathrm dt. $$
But on a pseudo-Riemannian manifold --- I have a 4-manifold with Lorentz signature in mind ---, can I define define a volume within the 'light cone' of an observer at point $p_0=(x_0,y_0,z_0,t_0)$.
It seems like if I limit the scope of my integral to the volume delimited by the light cone (I sum over the points $p_i$ where $d(p_0,p_i)>0$), and if I limit my change of coordinates to other observers within the light cone of $p_0$, then I should get most if not all of the properties of a volume --- such that it is always positive, etc. Is it a 'volume' in the traditional geometric sense?