nonmeasurable

Nonmeasurable is a term used in measure theory to describe a set or a function that is not measurable with respect to a given sigma-algebra or measure. In a measure space (X, Σ, μ), a subset E ⊆ X is measurable if E ∈ Σ. A function f: X → Y is measurable if the preimage of every measurable set in Y is in Σ.

A standard context is Lebesgue measure on the real numbers. The Lebesgue measurable sets form a sigma-algebra containing all open and closed sets, but not every subset of R is measurable. The existence of nonmeasurable sets is typically established using the axiom of choice. A famous example is a Vitali set, constructed within the interval [0,1], which cannot be assigned a Lebesgue measure consistently.

Nonmeasurable sets have implications for integration and probability. In the usual framework of probability theory, events are modeled as measurable sets; nonmeasurable events cannot be assigned a probability in the standard way. To demonstrate nonmeasurability, one can take the indicator function of a nonmeasurable set, which then fails to be measurable with respect to the given sigma-algebra.

Beyond sets, the term also applies to functions. A function can be nonmeasurable if it fails to have the property that the preimage of a measurable set is measurable. Nonmeasurable constructs often arise in theoretical discussions of measure theory, logic, and set theory, and their study highlights the limits of assigning sizes or probabilities to all subsets of a given space.