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.

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_i, t_i+1) of Π; suppose that this constant value is c_i ∈ 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 Π₁ is another partition of [a, b] 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.

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_n)_n∈N 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 : E → F with the same (finite) operator norm.

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