TITLE: In the space of probability distributions, is the set of discrete distributions dense?
QUESTION [5 upvotes]: Is the following true:

In the space of probability distributions, the set of discrete  distributions is dense with regard to the Lévy metric.

Can someone point me to any reference on this subject?

REPLY [3 votes]: There is a more general theorem in Parthasarathy's Probability measures on metric spaces on page 66:

Let $X$ be a separable metric space and $E\subset X$ dense in $X$. Then the set of all measures whose supports are finite subsets of $E$ is dense in $U(X)$.

In the book "measure" means probability measure, and $U(X)$ is the space of all probability measures.