multiinterval

A multiinterval is a subset of the real line that consists of a finite union of disjoint intervals. Formally, a multiinterval I is often written as I = ⋃_{k=1}^n [a_k, b_k], where n is a positive integer, and the endpoints satisfy a_1 < b_1 < a_2 < b_2 < … < a_n < b_n. Here the simplest case is a single interval [a_1, b_1], but the term specifically emphasizes the presence of multiple components. In many texts the constituent intervals may be closed, open, or half-open; when closed intervals are used, I is a compact set.

A multiinterval has several basic properties. It is compact (being a finite union of compact sets), and it has exactly n connected components, each of which is a closed interval. The Lebesgue measure of a multiinterval is the sum of the lengths of its components: m(I) = ∑_{k=1}^n (b_k − a_k). The complement of a multiinterval in the real line is a union of open intervals, along with potential unbounded ends if considered within the entire real line.

Examples include [0,1] ∪ [2,3] and [−5,−1] ∪ [0,1] ∪ [4,6].

In mathematical analysis, multiintervals arise in approximation theory, potential theory, and spectral theory, particularly in problems involving functions, measures, or polynomials supported on several disjoint intervals. They also appear in discussions of finite-gap potentials and in the study of rational approximation on unions of intervals.

See also: union of intervals, finite union, measure theory, potential theory.