@Victor Protsak: Done.

Well, actually $n+2$ $\delta$-functions are required for an arbitrary measurable space $X$ according to the theorem in my answer. A refined argument as in Noah's answer shows that when $X=[0,1]$ $n+1$ $\delta$-functions are enough.

The counterexample in Noah's answer shows that only $n$ Dirac measures may not be enough.

I don't understand how you obtained the geodesic equations. In general, they have the form $$\frac{d^2x^j}{dt^2}+\Gamma^{j}_{ik}\frac{dx^k}{dt}\frac{dx^i}{dt}=0$$ so the coefficients against the second derivatives should be equal to 1.