Editing Absolute convergence (section)

==Absolute convergence of integrals==

The [[integral]] <math display=inline>\int_A f(x)\,dx</math> of a real or complex-valued function is said to '''converge absolutely''' if <math display=inline>\int_A \left|f(x)\right|\,dx < \infty.</math>  One also says that <math>f</math> is '''absolutely integrable'''.  The issue of absolute integrability is intricate and depends on whether the [[Riemann integral|Riemann]], [[Lebesgue integral|Lebesgue]], or [[Henstock–Kurzweil integral|Kurzweil-Henstock]] (gauge) integral is considered; for the Riemann integral, it also depends on whether we only consider integrability in its proper sense (<math>f</math> and <math>A</math> both [[Bounded function|bounded]]), or permit the more general case of improper integrals.

As a standard property of the Riemann integral, when <math>A=[a,b]</math> is a bounded [[Interval (mathematics)|interval]], every [[continuous function]] is bounded and (Riemann) integrable, and since <math>f</math> continuous implies <math>|f|</math> continuous, every continuous function is absolutely integrable.  In fact, since <math>g\circ f</math> is Riemann integrable on <math>[a,b]</math> if <math>f</math> is (properly) integrable and <math>g</math> is continuous, it follows that <math>|f|=|\cdot|\circ f</math> is properly Riemann integrable if <math>f</math> is.  However, this implication does not hold in the case of improper integrals.  For instance, the function <math display=inline>f:[1,\infty) \to \R : x \mapsto \frac{\sin x}{x}</math> is improperly Riemann integrable on its unbounded domain, but it is not absolutely integrable:
<math display=block>\int_1^\infty \frac{\sin x}{x}\,dx = \frac{1}{2}\bigl[\pi - 2\,\mathrm{Si}(1)\bigr] \approx 0.62, \text{ but } \int_1^\infty \left|\frac{\sin x}{x}\right| dx = \infty.</math>
Indeed, more generally, given any series <math display=inline>\sum_{n=0}^\infty a_n</math> one can consider the associated [[step function]] <math>f_a: [0,\infty) \to \R</math> defined by <math>f_a([n,n+1)) = a_n.</math>  Then <math display=inline>\int_0^\infty f_a \, dx</math> converges absolutely, converges conditionally or diverges according to the corresponding behavior of <math display=inline>\sum_{n=0}^\infty a_n.</math>

The situation is different for the Lebesgue integral, which does not handle bounded and unbounded domains of integration separately (''see below'').  The fact that the integral of <math>|f|</math> is unbounded in the examples above implies that <math>f</math> is also not integrable in the Lebesgue sense.  In fact, in the Lebesgue theory of integration, given that <math>f</math> is [[Measurable function|measurable]], <math>f</math> is (Lebesgue) integrable if and only if <math>|f|</math> is (Lebesgue) integrable.  However, the hypothesis that <math>f</math> is measurable is crucial; it is not generally true that absolutely integrable functions on <math>[a,b]</math> are integrable (simply because they may fail to be measurable): let <math>S \subset [a,b]</math> be a nonmeasurable [[subset]] and consider <math>f = \chi_S - 1/2,</math> where <math>\chi_S</math> is the [[Indicator function|characteristic function]] of <math>S.</math>  Then <math>f</math> is not Lebesgue measurable and thus not integrable, but <math>|f| \equiv 1/2</math> is a constant function and clearly integrable.

On the other hand, a function <math>f</math> may be Kurzweil-Henstock integrable (gauge integrable) while <math>|f|</math> is not.  This includes the case of improperly Riemann integrable functions.

In a general sense, on any [[measure space]] <math>A,</math> the Lebesgue integral of a real-valued function is defined in terms of its positive and negative parts, so the facts:

# <math>f</math> integrable implies <math>|f|</math> integrable
# <math>f</math> measurable, <math>|f|</math> integrable implies <math>f</math> integrable

are essentially built into the definition of the Lebesgue integral.  In particular, applying the theory to the [[counting measure]] on a [[Set (mathematics)|set]] <math>S,</math> one recovers the notion of unordered summation of series developed by Moore–Smith using (what are now called) nets.  When <math>S = \N</math> 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 analysis|functional analytic]] approach, obtaining the [[Bochner integral]].