Background
Let $X \in \mathbb{R}^2$ be a random variable distributed over the set $D \subset \mathbb{R}^2$. The probability density function (PDF) of $X$ is given by:
\begin{equation*} p_X(x) = \frac{1_{D}(x)}{|D|_2} \end{equation*}
where \begin{equation*} 1_D(x)\triangleq \begin{cases} 1 & \text{if } x \in D, \ 0 & \text{otherwise} \end{cases} \qquad |D|_2\triangleq \textrm{Area}(D)=\int_D \text{d}x \end{equation*}
Given $p_X(x)$, the probability that a realization of $X$ is inside a region $S \subseteq \mathbb{R}^2$ is:
\begin{equation*} \mathbb{P}(X \in S) = \int_S p_X(x)\text{ d}x \end{equation*}
If I'm not mistaken, it can be expressed as:
\begin{equation*} \mathbb{P}(X \in S) = \int_S \frac{1_D(x)}{|D|_2}\text{ d}x= \frac{|S \cap D|_2}{|D|_2} \end{equation*}
My problem
Suppose that $D$ is a line connecting two vertices $V_1, V_2 \in \mathbb{R}^2$, i.e.,
\begin{equation*} D\triangleq {(1-\lambda)V_1+\lambda V_2 }_{\lambda\in[0,1]} \end{equation*}
then $|D|_2=0$, and $p_X(x)$ is undefined. However, it appears that the following formula still holds:
\begin{equation*} \mathbb{P}(X \in S)=\frac{|S\cap D|_1}{|D|_1} \end{equation*}
where \begin{equation*} |D|_1 \triangleq \textrm{Length}(D)= \lVert V_2 - V_1\rVert \qquad |S\cap D|_1 \triangleq \textrm{Length}(S\cap D) \end{equation*}
Therefore, I am wondering if it is somehow possible to define a PDF even in cases where $D$ has null area. I am considering expressing the PDF in terms of the Dirac delta, but I am unsure of how to proceed (if it is possible and I haven't made any mistakes).