Let [a, b] be a fixed closed, boundedinterval in the real lineR. 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 (ti, ti+1) of Π; suppose that this constant value is ci ∈ 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.
Extension to regulated functionsedit
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 (φn)n∈N such that || φn − f ||∞ → 0 as n → ∞; or, equivalently,
for all ε > 0, there exists a step function φε such that || φε − 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
exists, and, for every t ∈ (a, b], the left-sided limit
exists as well.
Define the integral of a regulated function f to be
where (φn)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 operatorT0 defined on a denselinear subspaceE0 of a normed linear spaceE and taking values in a Banach space F extends uniquely to a bounded linear operator T : E → F with the same (finite) operator norm.
Properties of the regulated integraledit
The integral is a linear operator: for any regulated functions f and g and constants α and β,
The integral is also a bounded operator: every regulated function f is bounded, and if m ≤ f(t) ≤ M for all t ∈ [a, b], then
In particular:
Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.
Extension to functions defined on the whole real lineedit
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:
the partition on whose open intervals a step function is required to be constant is allowed to be a countable set, but must be a discrete set, i.e. have no limit points;
the requirement of uniform convergence must be loosened to the requirement of uniform convergence on compact sets, i.e. closed and bounded intervals;
Berberian, S.K. (1979). "Regulated Functions: Bourbaki's Alternative to the Riemann Integral". The American Mathematical Monthly. 86 (3). Mathematical Association of America: 208. doi:10.2307/2321526. JSTOR 2321526.
Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.