# Labor Saving Device

As we move beyond the elementary probability courses in college and enter measure-theoretic probability, we learn that it is not the probability density function that is primitive to the concept of distributions, but rather the cumulative distribution function, $$\mathsf{P}(X \leq x)$$. It is not clear why it had to be of the form $$X \leq x$$ and not $$a < X < b$$ or $$X > x$$ or any set that we could possibly think of. Techniquely, we could choose $$a < X < b$$ or $$X > x$$ instead of $$X \leq x$$ to be the primitive set of interest. The real question is, what’s so special about these sets?

The general idea of a random variable starts with the fact that we can map an abstract space that represents the real world, $$(\Omega, \mathcal{M}, \mathsf{P})$$, to the real line, $$(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mathsf{P}_X)$$. That is, $X: (\Omega, \mathcal{M}, \mathsf{P}) \mapsto (\mathbb{R}, \mathcal{B}(\mathbb{R}), \mathsf{P}_X).$ And by the definition of a measurable function, which by the way random variable is, mandates that the inverse image of any set $$B \in \mathcal{B}(\mathbb{R})$$ to be in the $$\sigma$$-algebra $$\mathcal{M}$$, i.e., $$X^{-1}(B) \in \mathcal{M}$$. Thus, rigorously speaking, this means we need to examine every single Borel set $$B \in \mathcal{B}(\mathbb{R})$$ to verify if a function $$X$$ is measurable and thus is a random variable. I bet probabilists were looking for (unless this was too natural to even spend time thinking about) a way to avoid this complexity. Here comes the Labor-Saving Device!

(Labor-Saving Device) Suppose that $$X:\Omega\to \mathbb{R}$$ and $$\mathcal{C}\subseteq \mathcal{B}(\mathbb{R})$$ is such that $$\sigma(\mathcal{C}) = \mathcal{B}(\mathbb{R})$$. Further suppose that $$X^{-1}(C) \in \mathcal{M}$$ for every $$C \in \mathcal{C}$$. Then, $$X^{-1}(B) \in \mathcal{M}$$ for every $$B \in \mathcal{B}(\mathbb{R})$$.

According to this Labor-Saving Device, we can look at a subcollection of the Borel $$\sigma$$-algebra that generates Borel $$\sigma$$-algebra and check the sets in that subcollection. This property can be proven through the Good Sets Principle. Anyway, this Labor-Saving Device provides a way to focus on a smaller collection of sets than the Borel $$\sigma$$-algebra, and that is chosen to be the collection of $$(-\infty,x]$$ since $$\sigma((-\infty,x])=\mathcal{B}(\mathbb{R})$$.