Regulated integral

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

${\displaystyle \Pi =\{a=t_{0}<t_{1}<\cdots <t_{k}=b\}}$

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

${\displaystyle \int _{a}^{b}\varphi (t)\,\mathrm {d} t:=\sum _{i=0}^{k-1}c_{i}|t_{i+1}-t_{i}|.}$

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]:

• there is a sequence of step functions (φ_n)_n∈N such that || φ_n − f ||_∞ → 0 as n → ∞; or, equivalently,
• for all ε > 0, there exists a step function φ_ε such that || φ_ε − f ||_∞ < ε; or, equivalently,
• f lies in the closure of the space of step functions, where the closure is taken in the space of all bounded functions [a, b] → R and with respect to the supremum norm || - ||_∞; or equivalently,
• for every t ∈ [a, b), the right-sided limit

${\displaystyle f(t+)=\lim _{s\downarrow t}f(s)}$

exists, and, for every t ∈ (a, b], the left-sided limit

${\displaystyle f(t-)=\lim _{s\uparrow t}f(s)}$

exists as well.

Define the integral of a regulated function f to be

${\displaystyle \int _{a}^{b}f(t)\,\mathrm {d} t:=\lim _{n\to \infty }\int _{a}^{b}\varphi _{n}(t)\,\mathrm {d} t,}$

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.