Absolute Convergence - Absolute Convergence of Integrals

Absolute Convergence of Integrals

The integral of a real or complex-valued function is said to converge absolutely if One also says that f is absolutely integrable.

When A = is a closed bounded interval, every continuous function is integrable, and since f continuous implies |f| continuous, similarly every continuous function is absolutely integrable. It is not generally true that absolutely integrable functions on are integrable: let be a nonmeasurable subset and take, where is the characteristic function of S. Then f is not Lebesgue measurable but |f| is constant. However, it is a standard result that if f is Riemann integrable, so is |f|. This holds also for the Lebesgue integral; see below. On the other hand a function f may be Kurzweil-Henstock integrable (or "gauge integrable") while |f| is not. This includes the case of improperly Riemann integrable functions.

Similarly, when A is an interval of infinite length it is well known that there are improperly Riemann integrable functions f which are not absolutely integrable. Indeed, given any series one can consider the associated step function defined by . Then converges absolutely, converges conditionally or diverges according to the corresponding behavior of

Another example of a convergent but not absolutely convergent improper Riemann integral is .

On any measure space A the Lebesgue integral of a real-valued function is defined in terms of its positive and negative parts, so the facts:

1. f integrable implies |f| integrable
2. f measurable, |f| integrable implies f integrable

are essentially built into the definition of the Lebesgue integral. In particular, applying the theory to the counting measure on a set S, one recovers the notion of unordered summation of series developed by Moore-Smith using (what are now called) nets. When S = N is the set of natural numbers, Lebesgue integrability, unordered summability and absolute convergence all coincide.

Finally, all of the above holds for integrals with values in a Banach space. The definition of a Banach-valued Riemann integral is an evident modification of the usual one. For the Lebesgue integral one needs to circumvent the decomposition into positive and negative parts with Daniell's more functional analytic approach, obtaining the Bochner integral.