Measurable Functions

A random variable is really just a function from outcomes to numbers — a mapping from "which scenario happened" to "what value did we observe." But measure theory won't let you compute a probability like $P(X \le 5)$ for just any function $X$; it requires that the set of outcomes producing $X \le 5$ actually belongs to the collection of events you're allowed to assign probabilities to (the sigma-algebra). A function $X$ is called measurable precisely when this holds for every such threshold: the preimage of any measurable set in the output space is a measurable set in the input space.

Without measurability, an expression like $P(X \in A)$ is simply undefined — there's no guaranteed event in your probability space corresponding to "$X$ landed in $A$," so the question can't be asked, let alone answered. Measurability is the bridge that lets you translate statements about a function's output back into statements about the underlying sample space where probabilities actually live.

For example, on the real line with the standard Borel sigma-algebra, any continuous function is automatically measurable, because the preimage of an open interval under a continuous function is itself open — this is why virtually every function used in practice (payoffs, indicator functions, prices) is measurable without anyone needing to check by hand.

Measurability is a prerequisite you almost never see violated in applied work, but it's the technical scaffolding underneath every "let $X$ be a random variable" statement.