In classical measure theory, it is usually assumed that is the real line (or a variation such as the extended real line or the complex plane) equipped with the Borel sets. Then is measurable if and only if is measurable whenever is an interval. More generally, if is any topological space equipped with the Borel sets, then is measurable if and only if is measurable whenever is open.
In some variations of measure theory based on ‑ or -rings instead of on -algebras, it is necessary to allow partial functions whose domain is a relatively measurable set. Classically (when is the real line), one achieves (for purposes of integration) essentially the same result by requiring only that be measurable whenever is an interval that does not contain ; in other words, one effectively assumes that is zero wherever it would otherwise be undefined.
If (as in a measure space, a Cheng space, or a localisable measurable space), we have a notion of null sets (or full sets) in and , then we may allow a measurable function to be an almost function: a partial function whose domain is full. Specifically, an almost function is measurable if the preimage of every full set in is full in and the preimage of every measurable set in is, if not quite measurable in , at least equal to a measurable set in up to a full set in . (To emphasise this last change, we may call such functions almost measurable.) Additionally, we consider two measurable almost functions to be equal (or equivalent if one prefers) if they are almost equal: their equaliser is full.