From discrete to continuous distribution What happens to the indicator function of a simple event when we pass from discrete to continuous?
Assume to have a discrete collection of elements $\mathcal{X} = \{x_1, ..., x_N\} \subseteq [0,1]$ such that $\mathbb{P}[X = x_i] = p_\mathcal{X}(x_i)$. Consider the event $\mathcal{E} = \mathbb{1}_{\{X = x_i\}},$ where $\mathbb{1}$ is the indicator function.
If we consider a continuous set $\mathcal{X}$ covered by a continuous distribution $\bar{p}_\mathcal{X}$, since the event $\mathcal{E}$ has zero probability for all $x\in\mathcal{X}$ under the measure $\bar{p}_\mathcal{X}$, how can we write the event $\mathcal{E}$? Is there any formalism to describe this?
 A: The closest analog of the probability mass function $P\left(X=x\right)$ in the continuous case is the probability density function, which is interpreted as
$$f_X\left(x\right)dx = P\left(X \in \left[x,x+dx\right]\right)$$
where $dx$ is an infinitesimal positive value. Thus, $\int_x f_X\left(x\right)dx = 1$ by analogy with the pmf summing up to $1$ over all $x$.
The density is derived from the cumulative distribution function $P\left(X\leq x\right)$, so the "fundamental" class of events which is used to characterize the distribution is $\left\{X\leq x\right\}$ for all $x\in\mathbb{R}$. The probability of any such event is well-defined for any random variable, even if the density doesn't exist.
A: Usually people write $P[X \in [a, b]]$, where $[a, b] \subseteq \chi$ and that event is not necessarily of zero probability. What you call $\chi$ is referred to as "Sample space" (https://en.wikipedia.org/wiki/Sample_space) and is denoted by the letter $\Omega$. Also, the probability is calculated given an object from a $\sigma$-algebra (https://en.wikipedia.org/wiki/%CE%A3-algebra). To simplify things it's guaranteed all segments $[a, b]$ are in the Borel $\sigma$-algebra (which is used when you deal with continuous random variables). All the points $x \in \Omega$ are also in the Borel $\sigma$-algebra, that's why it's legal to compute probabilities of points and say they're equal to zero for continuous variables.    This formalism is called the Kolmogorov axioms (https://en.wikipedia.org/wiki/Probability_axioms).
