In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonné.


Definition on step functions

Let [a, b] be a fixed closed, bounded interval in the real line R. A real-valued function φ : [ab] → R is called a step function if there exists a finite partition

of [a, b] such that φ is constant on each open interval (ti, ti+1) of Π; suppose that this constant value is ciR. Then, define the integral of a step function φ to be

It can be shown that this definition is independent of the choice of partition, in that if Π1 is another partition of [ab] such that φ is constant on the open intervals of Π1, then the numerical value of the integral of φ is the same for Π1 as for Π.

Extension to regulated functions

A function f : [a, b] → R is called a regulated function if it is the uniform limit of a sequence of step functions on [a, b]:

Define the integral of a regulated function f to be

where (φn)nN is any sequence of step functions that converges uniformly to f.

One must check that this limit exists and is independent of the chosen sequence, but this is an immediate consequence of the continuous linear extension theorem of elementary functional analysis: a bounded linear operator T0 defined on a dense linear subspace E0 of a normed linear space E and taking values in a Banach space F extends uniquely to a bounded linear operator T : EF with the same (finite) operator norm.

Properties of the regulated integral

Extension to functions defined on the whole real line

It is possible to extend the definitions of step function and regulated function and the associated integrals to functions defined on the whole real line. However, care must be taken with certain technical points:

Extension to vector-valued functions

The above definitions go through mutatis mutandis in the case of functions taking values in a Banach space X.

See also