Regulated integral

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In mathematics, the regulated integral is a definition of integration for uniform limits of step functions. It 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] \subsetneq \mathbb{R} be a fixed interval. A function \phi : [a, b] \to \mathbb{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). Let

\phi|_{(t_{i}, t_{i + 1})} (t) \equiv c_{i}.

Define the integral of a step function φ to be

\int_{a}^{b} \phi(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.

[edit] Extension to regulated functions

A function f : [a, b] \to \mathbb{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 (\phi_{n})_{n = 1}^{\infty} such that \| \phi_{n} - f \|_{\infty} \to 0 as n \to \infty; or, equivalently,
  • for all \varepsilon > 0, there exists a step function \phi_{\varepsilon} such that \| \phi_{\varepsilon} - f \|_{\infty} < \varepsilon; or, equivalently,
  • f lies in the completion of the space of step functions with respect to the uniform norm \| - \|_{\infty}.

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} \phi_{n} (t) \, \mathrm{d} t,

where (\phi_{n})_{n = 1}^{\infty} 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.

[edit] Properties of the regulated integral

The integral is linear: for regulated functions f,g and constants α,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.

If f is a bounded function, with m \leq f(t) \leq M for all t \in [a, b], then

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.

Every continuous function on [a,b] is uniformly continuous, and hence regulated, so continuous functions have a regulated integral. Piecewise continuous functions are also integrable.

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 the 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] References

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

[edit] See also

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