Dirac Delta function Notation

Question

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is

$\pi(x_{1:n}) = \frac{1}{N}\sum^N_{i=1}\delta_{X^i_{1:n}}(x_{1:n})$

where $X^i_{1:n} \sim \pi(x_{1:n})$ for i=1,..,N

I'm wondering whether this is equivalent to the following

$\pi(x_{1:n}) = \frac{1}{N}\sum^N_{i=1}\delta(x_{1:n}-{X^i_{1:n}})$

for a specific value of $x_{1:n}$, say $x_{1:n} = y$. I understand the latter to mean that the probability of a particular $x_{1:n}$ is given by the number of samples that take this value, (hence $\delta(y - X^i_{1:n}) = \delta(0) = 1 )$ divided by the total number of samples, N.

Is this correct?

I realise my questions is similar to the question asked here: Delta Function in Monte Carlo Sampling and have read it and looked at linked answers.

Answer 1

Yeah, that's correct. You can see from the wikipedia definition here how the generalization with subscript is defined, and your transformation, in the case at hand, renders it back into original delta notation.