I'm a PhD student working on stochastic optimization. In the course of my work, I happen to meet measure theory, which is far from being my cup of thea.
Especially, I need to know the answer to the following problem:

Let P(R^n) the set of Borel probability measures on R^n.

Let D(R^n) C P(R^n) the set of DISCRETE probability measures having a
positive weight on a FINITE number of vectors.

Finally, consider the metric space (P_1(R^n), W_1)
(Wasserstein metric - Wikipedia, the free encyclopedia).

Is D(R^n) dense in (P_1(R^n), W_1) ?

Any hint to answer that question is welcome ! .

Thank you in advance.