I'm interested in cases where you have a probability function $\Pr$ defined over the values of the continuous real valued variable $X$, where some particular value or values have non-zero, non-unitary probability, and the remaining probability is distributed over intervals.
I'm inclined to deal with such cases by combining a density function $f$, defined over all values of $X$ and a mass function $m$ defined over some countable subset of values of $X$, $\mathbf{x} = \{ x_1, x_2, \dots \}$, such that $$ \int_{X} f(x)\hspace{4pt} dx \quad +\quad \sum_{x_i \in \mathbf{x}} m(x_i) = 1 $$ and then defining $$ \Pr(X \in [a, b]) = \int_a^b f(x) \hspace{4pt}dx \quad + \quad \sum_{\substack{ x_i \in \mathbf{x}: \\ x_i \in [a, b]}} m(x_i) $$
However, I don't see such mixed density/mass distributions discussed in any of the texts I usually consult on probability theory. Is there a standard way of dealing with such problems other than this? Or are there issues for this way of proceeding? Any guidance would be much appreciated.
