Integrable function

In mathematics, an integrable function is a function whose integral exists. Unless specifically stated, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann-integrable" (i.e., its Riemann integral exists), "Henstock-Kurzweil-integrable," etc.

Notice that a function may have an indefinite integral (antiderivative), and yet not be integrable as defined below. For instance the function

$F(x)=\sin(x)$

is the antiderivative of

$f(x)=\cos(x)$

but f(x) is not considered an integrable function over the real numbers. This may be the case even if the antiderivative has a limit in each direction, like

$F(x)=\sin(x)/x$ $(x\geq 1)$

whose derivative, $f(x)=\cos(x)/x-\sin(x)/x^{2}$, is not integrable from 1 to infinity. This is true even if the interval of integration is not infinite, as with the antiderivative

$F(x)=x\sin(1/x)$ $(0<x\leq 1)$

whose derivative $f(x)=\sin(1/x)-\cos(1/x)/x$ is not integrable from 0 to 1. (Whatever value is assigned to f(x) at 0, it will be discontinuous there, and F'(0) is not defined, so the Corollary of the Fundamental Theorem of Calculus does not apply to the interval [0, 1].)

Lebesgue integrability

Given a set X with sigma-algebra σ defined on X and a measure μ on σ, a real-valued function f:X → R is integrable if both the positive part f ⁺ and the negative part f ⁻ are measurable functions whose Lebesgue integral is finite. Let

$f^{+}=\max(f,0),\ f^{-}=\max(-f,0)$

be the "positive" and "negative" part of f. If f is integrable, then its integral is defined as

$\int f=\int f^{+}-\int f^{-}.$

For a real number p ≥ 0, the function f is p-integrable if the function | f | ^p is integrable; for p = 1 one says absolutely integrable. (Notice that f(x) is integrable if and only if |f(x)| is integrable, so the terms "integrable" and "absolutely integrable" are really the same thing.) The term p-summable is sometimes used as well, especially if the function f is a sequence and μ is discrete.

The L ^p spaces are one of the main objects of study of functional analysis.

Square-integrable

A real- or complex-valued function of a real or complex variable is square-integrable on an interval if the integral of the square of its absolute value, over that interval, is finite. The set of all measurable functions that are square-integrable, in the sense of the Lebesgue integral, forms a vector space which is a Hilbert space, the so-called L² space, provided functions which are equal almost everywhere are identified. (Formally, L² is the quotient space of the space of square integrable functions by the subspace of functions which vanish almost everywhere.)

This is especially useful in quantum mechanics as wave functions must be square integrable over all space if a physically possible solution is to be obtained from the theory.