Generalised probability density functions using Dirac deltas

Question

I just cannot understand how a discrete random variable $X$ may be represented using a generalised probability density function (p.d.f.) $f_X$ as,

\begin{equation}\tag{1} f_X(x) = \sum_{x_k \in R_X} P(X = x_k)\delta(x - x_k) \end{equation} where $R_X = \{x_1, x_2, ...\}$ is the range of the discrete random variable $X$, $P(\cdot)$ is the probability mass function (p.m.f.) of $X$ and $\delta(x - x_k)$ is the Dirac delta function,

$$\delta(x) = \begin{cases} \infty,& x = 0\\ 0, & otherwise.\end{cases} $$

where by definition of the Dirac delta function,

$$\int_{- \infty}^{+ \infty} \delta(x)dx = 1$$

I have seen (1) used in a number of online sources. What I do not understand is how (1) works if you end up multiplying infinities? Shouldn't (1) be rewritten as,

\begin{equation}\tag{2} f_X(x) = \sum_{x_k \in R_X} P(X = x_k)\delta(x - x_k)dx \end{equation}

where,

$$\delta(x)dx = \begin{cases} 1,& x = 0\\ 0, & otherwise.\end{cases} $$

Two online sources where (1) is used:

i. A book's supporting website

ii. A tutorial publication (pdf)

Answer 1

The point here is that $f_X(x)$ is a probability density function, so it's only ever used to calculate anything in an integral. That takes care of the issue with the Dirac delta function. Basically this is a way to turn a discrete probability distribution $P(X)$ into a continuous one $f_X(x)$.