Measurability

Measurability is the property of a quantity, object, or set that makes it possible to assign a value or classify it according to a defined measurement framework. In mathematics, measurability is formalized within measure theory and probability, enabling rigorous treatment of sizes, probabilities, and integrals.

In measure theory, a measure space consists of a set X, a sigma-algebra F of subsets of X, and a measure μ that assigns sizes to sets in F. A subset of X is called measurable if it belongs to F. A function f: X → R (or extended reals) is measurable if the preimage of every Borel set in R is in F; equivalently, the function preserves measurability with respect to the sigma-algebras on its domain and codomain. This framework underpins the construction of integrals and the formal development of probability theory. Common examples include Lebesgue measurable sets on the real line and Borel measurable functions.

In probability theory, a random variable is a measurable function from a probability space to the real numbers (or another measurable space). The measurability condition ensures that the events {X ≤ t} and similar sets are events with well-defined probabilities, allowing the use of expectation, variance, and other statistical tools.

In the applied sciences, measurability is tied to the capabilities of measurement instruments, calibration, and the specification of measurement scales and units. Some quantities may be only partially measurable or require indirect inference, and measurement uncertainty and errors are intrinsic considerations in practical contexts.

Overall, measurability links theoretical constructs to observable quantities by specifying what can be meaningfully measured within a given framework.