A σ-algebra hands us a rich family of sets we are allowed to measure. But the whole point of building that machinery was never sets for their own sake — it was to integrate functions. So the question that organises this entire page is simple: given a measure, which functions can we integrate?

Recall Lebesgue's strategy — slice the range, not the domain. Instead of asking "what is f here?" strip by strip along the x-axis, we sort by value: for each height y we ask where is f about this high, and weight that level by the size of the set of x that reach it:

For that formula to even make sense, every one of those level sets \{x : f(x) > y\} must be a set we can measure — a member of our σ-algebra. A function all of whose level sets are measurable is exactly the kind of function this recipe can weigh. These are the measurable functions, and they are the precise analogue, for measure theory, of the continuous functions of ordinary analysis. This page defines them, proves they are the robust class they look like, and ends at the bridge — simple functions — that carries us to the Lebesgue integral itself.

The definition — preimages land back in the σ-algebra

Fix two measurable spaces (X, \mathcal{F}) and (Y, \mathcal{G}) — a set with a distinguished σ-algebra of "measurable" subsets on each. A map f : X \to Y is (\mathcal{F}, \mathcal{G})-measurable if the preimage of every measurable set is measurable:

The overwhelmingly common case is a real-valued function f : X \to \mathbb{R} with the target carrying its Borel σ-algebra \mathcal{B}(\mathbb{R}) — the smallest σ-algebra containing every interval. Checking f^{-1}(B) \in \mathcal{F} for all Borel B sounds hopeless: there are enormously many Borel sets. The saving grace is that we only ever need to test a generating family. The rays (a, \infty) generate \mathcal{B}(\mathbb{R}), so the whole condition collapses to a single, checkable one:

The four equivalent criteria

In fact any one of four ray-conditions will do — and this is where the closure axioms of a σ-algebra earn their keep. Each of the following is equivalent to f being measurable:

The equivalences are pure countable bookkeeping. Complements swap strict for non-strict the easy way — \{f \le a\} = X \setminus \{f > a\} and \{f < a\} = X \setminus \{f \ge a\} — using closure under complement. To turn > into \ge we squeeze with a countable intersection, and back again with a countable union:

If f(x) \ge a then f(x) > a - \tfrac1n for every n; conversely if f(x) > a - \tfrac1n for all n, letting n \to \infty forces f(x) \ge a — so the two sides really are equal. Because a σ-algebra is closed under countable unions, intersections, and complements, each set on the right lies in \mathcal{F} whenever the pieces do. All four criteria stand or fall together. This is precisely the payoff of insisting on countable operations back when we built the σ-algebra: it is exactly enough to pass limits through.

The analogy that names the concept: measurable is to measure as continuous is to topology

The definition is a deliberate echo. Recall the topological characterisation of continuity: a map is continuous exactly when the preimage of every open set is open. Swap "open" for "measurable" and you have measurability verbatim:

This is not a loose metaphor; it has teeth. Since the open sets generate the Borel σ-algebra, a map that pulls open sets back to open sets certainly pulls Borel sets back to Borel sets. Hence:

Every polynomial, every trig function, every exponential you ever met is automatically measurable — you never have to check. And just as in topology you test continuity on a base rather than on all open sets, here you test measurability on a generator (the rays) rather than on all Borel sets. The difference — and it is the whole difference — is that the class of measurable functions is far larger and far more robust than the continuous ones, as we see next.

Closure: the class survives every operation analysis needs

Measurable functions form a class closed under essentially everything. If f, g : X \to \mathbb{R} are measurable and c \in \mathbb{R}, then so are

The proofs are variations on one trick — write the level set of the combination as a countable combination of level sets of the pieces. For the sum, a rational q can always be slipped between f(x) and a - g(x) whenever f(x) + g(x) > a, so

a countable union of measurable sets — measurable. (This is why the rationals' countability keeps mattering.)

The property that continuity lacks: closure under pointwise limits

Here is the decisive advantage. Let f_1, f_2, f_3, \dots all be measurable. Then all of the following are measurable too:

The keystone is the supremum, and it is a one-line miracle: a supremum exceeds a exactly when at least one term does, so its level set is a plain countable union —

The infimum follows by \inf_n f_n = -\sup_n(-f_n); then \limsup_n f_n = \inf_k \sup_{n \ge k} f_n and its mirror are just nested sups and infs; and when the limit exists it equals the limsup, so it is measurable as well.

Contrast this with continuity. A pointwise limit of continuous functions is famously not continuous — the sawtooth limits of Fourier analysis, or x^n \to \mathbf{1}_{\{1\}} on [0,1], are the standard cautionary tales. Measurability has no such fragility: you can take all the pointwise limits you like and never leave the class. This single fact is the structural reason the Lebesgue theory's convergence theorems (monotone convergence, dominated convergence) even have a class to live in — the limit of integrable functions is still a function we are allowed to integrate.

Simple functions and the approximation that builds the integral

A simple function is a finite combination of indicators of measurable sets:

It takes only finitely many values, and it is measurable — its level set \{\varphi > a\} is just the union of those A_k whose value c_k exceeds a, a finite (hence measurable) union. Simple functions are the ones we can integrate by hand: the Lebesgue integral of \varphi is simply \sum_k c_k\,\mu(A_k) — weight each height by the measure of the set on which it is achieved. They are the atoms of the whole theory.

And every non-negative measurable function is a limit of them — climbing up from below. This is the bridge to the next page:

The construction is explicit and dyadic. At level n, chop the range [0, n) into 2^n equal slabs of height 2^{-n}, round each value down to the slab beneath it, and cap the whole thing at height n:

Each \varphi_n is simple — the flooring makes it constant on the measurable level bands \{k\,2^{-n} \le f < (k+1)\,2^{-n}\}. Refining n \to n+1 halves every slab, so the staircase can only rise (\varphi_n \le \varphi_{n+1}), and at every point the rounding error is at most 2^{-n} \to 0 once f(x) < n, so \varphi_n(x) \to f(x). Watch it happen — drag the level up and the steps crowd toward the curve from underneath:

This one picture is the plan for the Lebesgue integral: define \int \varphi_n for the simple staircases (easy — a finite weighted sum of measures), then define \int f = \lim_n \int \varphi_n. Because the \varphi_n increase, that limit always exists in [0, \infty], and measurability is exactly what guarantees the level bands it rests on are things we can measure. Measurable functions are, quite literally, the functions this procedure can integrate.