measurable

Measurable is an adjective describing something that can be measured or quantified. In everyday use, a measurable quantity can be determined by observation or by a measuring instrument, and it is typically assigned a numerical value with an agreed unit and scale. The term also implies that the value is subject to some uncertainty or error, and that instruments or procedures exist to compare the result to a standard.

In mathematics, measurable has a specialized meaning in measure theory. A measurable space consists of a set equipped with a sigma-algebra of subsets, called measurable sets. A function between two measurable spaces is called measurable if the preimage of every measurable set is measurable. This condition ensures that operations such as integration are well defined. Common examples include Lebesgue measurable sets and Borel measurable functions. Measurable functions form the foundation of probability theory and real analysis, enabling the definition of expectations, distributions, and almost everywhere statements. A simple example is a constant function, which is measurable; the indicator function of a set is also measurable.

In statistics and probability, random variables are modeled as measurable functions on a probability space, ensuring that their distribution is well defined. The concept is central to convergence theorems, integration, and numerical methods.

In data collection and metrology, measurability relates to the ability to produce reproducible results with documented units, calibrations, and traceability to standards.