(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a standard way to generalize this integral to more general surfaces (rectifiable varifolds, perhaps)?

(b) For the special case of oriented piecewise linear (PL) surfaces without boundary, there is a formula in the literature for the total mean curvature, given by
 $$\sum_e {\rm length}(e) \,\theta(e),$$ where the sum is over all the edges and $\theta(e)\in (-\pi, \pi)$ is the signed angle between the normals to the adjacent faces at $e$. It is said that this formula is 'exact'. I cannot trace the origin of this formula, and cannot figure out in what sense is this formula exact.

It seems like this 'exactness' in (b) hinges on a definite answer to (a).

p.s. We do have a heuristical argument suggesting that the following is true:
For any sequence of smooth surfaces converging to a PL surface in a reasonable sense, the smooth total mean curvatures converge to the sum above. 

Thanks.