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 \(\sigma((-\infty,x])=\mathcal{B}(\mathbb{R})\).

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