I am confused by the dirac delta function in the formulation of monte carlo sampling. http://www.cs.ubc.ca/~arnaud/doucet_johansen_tutorialPF.pdf, for instance, defines in section 3.1 page 8 the marginal as $\pi(x_k)=\frac{1}{N}\sum_{i=1}^N\delta_{X^i_k}(x_k)$.

What does it mean here? That the density of $x_k$ is the average of the densities of the samples equal to $x_k$?

Thanks in advance.