Measurable Functions

Preimages of Borel sets, and a class that survives pointwise limits.

The idea

The measurable functions are the functions measure theory integrates: a class wide enough to contain every continuous function and stable under the pointwise limits that defeat the Riemann integral.

A $\sigma$-algebra on a set $X$ is a family of subsets containing $X$ and closed under complements and countable unions; its members are the measurable sets. The Borel sets of $\mathbb{R}$ are the smallest $\sigma$-algebra containing every open set.

A function $f \colon X \to \mathbb{R}$ is measurable when the preimage of every Borel set is measurable: $f^{-1}(B) = \{\, x \in X : f(x) \in B \,\} \text{ is measurable for every Borel } B.$ In practice one uses the equivalent workhorse test: $f$ is measurable if and only if $\{x : f(x) > a\}$ is measurable for every real $a$.

The definition has the same shape as continuity, which asks that the preimage of every open set be open: continuity pulls back the open sets, measurability pulls back the Borel sets. On $\mathbb{R}$ with the Borel sets as its measurable sets, a continuous map pulls every open set back to an open, hence Borel, set; since the open sets generate the Borel $\sigma$-algebra, this already forces every Borel set to pull back to a Borel set, so every continuous function is measurable.

The definition pulls sets back rather than pushing them forward because preimages commute with the operations a $\sigma$-algebra is closed under: the preimage of a complement is the complement of the preimage, and the preimage of a countable union is the union of the preimages. Images obey neither rule.

That is why the class is stable. Given a sequence of measurable functions $f_{1}, f_{2}, \dots$, the level sets of $\sup_{n} f_{n}$, $\inf_{n} f_{n}$, $\limsup_{n} f_{n}$ and $\liminf_{n} f_{n}$ are built from the level sets of the terms by countably many unions and intersections, so all four are measurable, and with them every pointwise limit. Over an uncountable family the argument fails, because a $\sigma$-algebra guarantees nothing about uncountable unions.

Ways to work on it

• Walkthrough. The preimage definition, the f > a test, and the closure properties.
• Proof. Why checking the rays f > a certifies every Borel preimage.
• Practice. Decide whether a construction stays inside the measurable class.
• Hardest. Build simple functions and use them to approximate any measurable function.

Not sure where to start? Take the ten-question placement test.