**EDIT:** Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov.

**Questions 1.** Can one define a measure $K$ on $S$ (thought to be an analogue of the Gauss curvature) satisfying the following properties:

(a) if the metric on $S$ is smooth then $K$ is the usual Gauss curvature times the Lebesgue measure.

(b) If a sequence of orientable surfaces $S_i$ with such metrics (with the same lower bound $\kappa$ on the curvatures) converges to $S$ in the Gromov-Hausdorff sense then $K_i\to K$ weakly (what is weak convergence of measures on different spaces should be made more precise, but I guess it is well known to experts).

(c) Gauss-Bonnet formula: $\int_S K=2\pi \chi(S)$.

(It is very likely that (c) follows from (a)+(b).)

**Question 2.** If the answer to Question 1 is positive, it seems likely that if $S$ has non-negative curvature which in some open subset is $\geq \kappa>0$ (and $S$ is orientable) then $S$ is homeomorphic to the 2-sphere. Is this consequence known to be true in the context of Alexandrov spaces? (For smooth Riemannian metrics it is well known.)

**UPDATE:**. The answer to both questions is YES as follows from the answer below by Thomas Richard and the comment by Anton Petrunin.