Regulated integral
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. Definition on step functions. Let ["a", "b"] be a fixed closed, bounded interval in the real line R. A real-valued function "φ" : ["a", "b"] → R is called a step function if there exists a finite partition of ["a", "b"] such that "φ" is constant on each open interval ("t", "t") of Π ; suppose that this constant value is "c" ∈ R. 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 Π is another partition of ["a", "b"] such that "φ" is constant on the open intervals of Π, then the numerical value of the integral of "φ" is the same for Π 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 ("φ") 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 "T" defined on a dense linear subspace "E" of a normed linear space "E" and taking values in a Banach space "F" extends uniquely to a bounded linear operator "T" : "E" → "F" with the same (finite) operator norm. 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 normed vector space "X".