TITLE: Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?
QUESTION [39 upvotes]: First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ is the Gaussian curvature, $dA$ is the area form and $\chi(M)$ is the Euler characteristic.
Theorem 2 (combinatorial version): Let $M$ be a two dimensional simplicial complex, with vertices, edges and faces (essentially a bunch of triangles glued together along the edges). Define the function $K:M \rightarrow \mathbb{R}$ to be zero at any point that is not a vertex. At any vertex sum up all the angles 
at the vertex and look at the deviation from $2 \pi$. This is the value of $K$ at a vertex. Then
$$ \sum_{p\in M} \frac{K(p)}{2 \pi} = \chi(M) . $$
Both these statements are known as Gauss Bonnet Theorem. My question is the following: Is it possible to use Theorem 2 in some way to prove Theorem 1? In other words can one recover Theorem 1 from Theorem 2 as some sort of an appropriate "limit"?
A second question is: Is there are a more general theorem from 
which both theorem 1 and theorem 2 arise as special cases? Probably 
some version of the theorem where $K$ simply has to be a measurable 
function?

REPLY [6 votes]: That should just be a comment but it's getting to long.
Also related is the theory of surfaces with bounded integral curvature developed in the 50's by A.D. Alexandrov and its followers. This class of surfaces contains both smooth and polyhedral surfaces.
To be more precise, in Alexandrov words a surface with bounded (integral) curvature is a compact topological surface $(X,d)$ with a geodesic distance $d$ such that for any finite collection of (nice) disjoint geodesic triangles $T_i$, $\sum |e(T_i)|<C$ for some $C$ indecent of the collection of triangles. Here the excess $e(T)$ of a triangle is "sum of angles of $T$ minus $2\pi$". (Actually I put quite a lot of stuff under the rug, you have to define what angles are in this context, and it's not clear they exist, so one consider upper angles instead).
From this one defines, essentially in the way Liviu did, a curvature measure $\omega$ on $M$, wich will be $K_gdv_g$ in the smooth case and $\sum K(p)\delta_p$ in the polyhedral case.
The two versions of Gauss Bonnet you gave can be rephrased as $\omega(M)=2\pi\chi(M)$, which in fact holds for all Alexandrov surfaces.
Moreover, one can deduce each one from the other by an approximation process. One can approach a smooth metric by a sequence of polyhedral metrics or a polyhedral metric by a sequence of smooth metrics in such a way such that the curvature measures with respect the approximating metrics (weakly) converge to the curvature measure of the approximated metric. 
This material can be found in 'The intrinsic geometry of surfaces with bounded curvature' by A.D. Aleksandrov and Zalgaller.