Regulated integral

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In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The regulated integral may be seen as a simple prototype for the Lebesgue integral in which the "simple functions" are the piecewise constant step functions. The use of the regulated integral instead of the Riemann integral was advocated by Nicolas Bourbaki.

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[edit] Definition

[edit] 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

\Pi = \{ a = t_{0} < t_{1} < \dots < t_{k} = b \}

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

\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.

[edit] 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]:

  • 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 || - ||∞.

Define the integral of a regulated function f to be

\int_{a}^{b} f(t) \, \mathrm{d} t := \lim_{n \to \infty} \int_{a}^{b} \varphi_{n} (t) \, \mathrm{d} t,

where (φn)n∈N is any sequence of step functions that converges uniformly to f. It can be shown that the definition of the integral is independent of the sequence of step functions chosen.

This definition and the fact that it is well-defined may be seen as a corollary of the continuous linear extension theorem that a bounded linear operator T0 defined on a dense linear subspace E0 of a Banach space E and taking values in another Banach space F extends uniquely to a bounded linear operator T : E → F with the same (finite) operator norm.

[edit] Properties of the regulated integral

  • The integral is a linear operator: for any regulated functions f and g and constants α and β,
\int_{a}^{b} \alpha f(t) + \beta g(t) \, \mathrm{d} t = \alpha \int_{a}^{b} f(t) \, \mathrm{d} t + \beta \int_{a}^{b} g(t) \, \mathrm{d} t.
m | b - a | \leq \int_{a}^{b} f(t) \, \mathrm{d} t \leq M | b - a |.
In particular:
\left| \int_{a}^{b} f(t) \, \mathrm{d} t \right| \leq \int_{a}^{b} | f(t) | \, \mathrm{d} t.
  • The regulated integral is a special case of the Lebesgue integral: every regulated function is Lebesgue integrable, and the two integrals agree.
  • The regulated and Riemann integrals also agree when both are defined.

[edit] 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:

  • 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;
  • not every bounded function is integrable (e.g. the function with constant value 1). This leads to a notion of local integrability.

[edit] 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.

[edit] References

  • Berberian, S.K. (1979). "Regulated Functions: Bourbaki's Alternative to the Riemann Integral". The American Mathematical Monthly. 
  • 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. 

[edit] See also

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