Regulated function

From Wikipedia, the free encyclopedia

In mathematics, a regulated function (or ruled function) is a "well-behaved" function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations.

[edit] Definition

Let X be a Banach space with norm || - ||_X. A function f : [0, T] → X is said to be a regulated function if one (and hence both) of the following two equivalent conditions holds true (Dieudonné 1969, §7.6):

• for every t in the interval [0, T], both the left and right limits f(t−) and f(t+) exist in X (apart from, obviously, f(0−) and f(T+));

• there exists a sequence of step functions φ_n : [0, T] → X converging uniformly to f (i.e. with respect to the supremum norm || - ||_∞).

It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:

• for every δ > 0, there is some step function φ_δ : [0, T] → X such that

• f lies in the closure of the space Step([0, T]; X) of all step functions from [0, T] into X (taking closure with respect to the supremum norm in the space B([0, T]; X) of all bounded functions from [0, T] into X).

[edit] Properties of regulated functions

Let Reg([0, T]; X) denote the set of all regulated functions f : [0, T] → X.

• Sums and scalar multiples of regulated functions are again regulated functions. In other words, Reg([0, T]; X) is a vector space over the same field K as the space X; typically, K will be the real or complex numbers. If X is equipped with an operation of multiplication, then products of regulated functions are again regulated functions. In other words, if X is a K-algebra, then so is Reg([0, T]; X).

• The supremum norm is a norm on Reg([0, T]; X), and Reg([0, T]; X) is a topological vector space with respect to the topology induced by the supremum norm.

• As noted above, Reg([0, T]; X) is the closure in B([0, T]; X) of Step([0, T]; X) with respect to the supremum norm.

• If X is a Banach space, then Reg([0, T]; X) is also a Banach space with respect to the supremum norm.

• Reg([0, T]; R) forms an infinite-dimensional real Banach algebra: finite linear combinations and products of regulated functions are again regulated functions.

• Since a continuous function defined on a compact space (such as [0, T]) is automatically uniformly continuous, every continuous function f : [0, T] → X is also regulated. In fact, with respect to the supremum norm, the space C⁰([0, T]; X) of continuous functions is a closed linear subspace of Reg([0, T]; X).

• If X is a Banach space, then the space BV([0, T]; X) of functions of bounded variation forms a dense linear subspace of Reg([0, T]; X):

• If X is a Banach space, then a function f : [0, T] → X is regulated if and only if it is of bounded φ-variation for some φ:

• If X is a separable Hilbert space, then Reg([0, T]; X) satisfies a compactness theorem known as the Fraňková-Helly selection theorem.

• The integral, as defined on step functions in the obvious way, extends naturally to Reg([0, T]; X) by defining the integral of a regulated function to be the limit of the integrals of any sequence of step functions converging uniformly to it. This extension is well-defined and satisfies all of the usual properties of an integral. In particular, the regulated integral
  • is a bounded linear function from Reg([0, T]; X) to X; hence, in the case X = R, the integral is an element of the space that is dual to Reg([0, T]; R);
  • agrees with the Riemann integral whenever both are defined.

[edit] References

• Dieudonné, Jean (1969), Foundations of modern analysis, Academic Press .
• Fraňková, Dana (1991), “Regulated functions”, Math. Bohem. 116 (1): 20-59 .
• 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 .
• Lang, Serge (1972), Differential manifolds, Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc. .