A function often appears before the measure does. We may write down a temperature field, a random payoff, a candidate density, or a solution of a differential equation before asking whether it can be integrated. The first obstruction is not size, continuity, or differentiability. It is observability: if we ask whether the value of the function lies in a set we can observe in the codomain, does that question correspond to a set we can observe in the domain?

This is the reason measurability is phrased through inverse images. A function $f: E \to G$ transports questions backward: an event $A \subset G$ becomes the event $f^{-1}(A) \subset E$. The function is measurable exactly when every observable question about values has an observable answer on the source.

[example: A Function That Cannot Be Integrated] Let $V \subset [0,1]$ be non-Lebesgue-measurable, and define $f:[0,1]\to\mathbb{R}$ by $f=\mathbb{1}_V$. Thus, for each $x\in[0,1]$, \begin{align*} f(x)=1 \text{ if } x\in V, \qquad f(x)=0 \text{ if } x\notin V. \end{align*} We compute the threshold set at level $1/2$. If $x\in V$, then $f(x)=1>1/2$, so $x\in\{y\in[0,1]:f(y)>1/2\}$. If $x\notin V$, then $f(x)=0\le 1/2$, so $x\notin\{y\in[0,1]:f(y)>1/2\}$. Therefore the two inclusions give \begin{align*} \{x\in[0,1]:f(x)>1/2\}=V. \end{align*} When $[0,1]$ is equipped with the Lebesgue measurable sets, this level set is not measurable because $V$ was chosen to be non-Lebesgue-measurable. Hence $f$ is not a permissible Lebesgue integrand: the basic value-question “is $f$ larger than $1/2$?” pulls back to a set outside the measurable structure on the domain. [/example]

The example shows that measurability is not a cosmetic condition added after a function is already understood. It decides whether the function belongs to the measure-theoretic universe at all. From this point of view, [measurable functions](/page/Measurable%20Functions) are the morphisms of measure theory: they preserve the structure that matters.

## Definition

### Measurable Structures

Before defining measurable functions, we need to name the measurable structure that lives on a set. The relevant structure is not a measure yet, but a collection of subsets stable under the set operations that countable limiting arguments require.

[definition: Sigma-Algebra] Let $E$ be a set. A sigma-algebra on $E$ is a collection $\mathcal{E} \subset \mathcal{P}(E)$ such that $E \in \mathcal{E}$, such that $E \setminus A \in \mathcal{E}$ whenever $A \in \mathcal{E}$, and such that $\bigcup_{n=1}^{\infty} A_n \in \mathcal{E}$ whenever $A_n \in \mathcal{E}$ for every $n \in \mathbb{N}$. [/definition]

A sigma-algebra by itself is only a list of admissible subsets; it does not name the ambient set as part of the object. Since measurability depends on both the points being mapped and the subsets allowed to be observed, the basic domain for measure theory must keep these two pieces together.

This becomes important whenever the same set carries different sigma-algebras, because the identity map may be measurable for one choice of observable sets and not for another. To make later definitions unambiguous, we need a single object that records both the underlying set and the sigma-algebra of observable subsets on it.

[definition: Measurable Space] A measurable space is a pair $(E, \mathcal{E})$ where $E$ is a set and $\mathcal{E}$ is a sigma-algebra on $E$. [/definition]

### The Inverse-Image Condition

With measurable spaces defined, the central problem is how a function should respect their observable sets. A function $f:E\to G$ transports questions backward: a set $A\subset G$ of observable values becomes the inverse image $f^{-1}(A)\subset E$. The function is admissible precisely when every observable value-question in the target pulls back to an observable source-event.

[definition: Measurable Function] Let $(E, \mathcal{E})$ and $(G, \mathcal{G})$ be measurable spaces. A function $f: E \to G$ is $\mathcal{E}/\mathcal{G}$-measurable if $f^{-1}(A) \in \mathcal{E}$ for every $A \in \mathcal{G}$. [/definition]

When the sigma-algebras are understood, we simply say that $f$ is measurable. The notation $\mathcal{E}/\mathcal{G}$ is useful when several sigma-algebras live on the same underlying sets, since the same pointwise function may be measurable for one choice and not for another. Thus the definition of measurable function says exactly that a map preserves observable value-events by pulling them back to observable source-events.

### Borel and Probabilistic Targets

The most common codomain in analysis is $\mathbb{R}$, not with every subset declared measurable, but with the sigma-algebra generated by open sets. The next definition is needed because it turns a [topological space](/page/Topological%20Space) into a measurable space without discarding the open-set information used in continuity.

[definition: Borel Sigma-Algebra] Let $(X, \tau)$ be a topological space. The Borel sigma-algebra on $X$, denoted $\mathcal{B}(X)$, is the smallest sigma-algebra on $X$ containing every [open set](/page/Open%20Set) in $\tau$. [/definition]

Topology often supplies the observable sets in the codomain: open sets are the primitive data, and the Borel sigma-algebra records all measurable events forced by them. Thus a function into a topological space can be tested for measurability without choosing a separate sigma-algebra by hand.

The resulting notion is especially useful because it includes continuous maps but also many discontinuous maps whose level events are still observable. This motivates a named class of maps whose source has measurable structure and whose target is observed through its Borel sets.