Integrable function

From Free net encyclopedia

In mathematics, the term integrable function refers to 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. Below we will only examine the concept of 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 or if both f ⁺ and f ^- are measurable functions with finite Lebesgue integral. Let

<math>

f^+ = \max (f,0) </math>

and

<math>

f^- = \max(-f,0) </math>

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

<math>

\int f = \mu(f^+ ) - \mu(f^- ) </math>

For a real number p ≥ 0, the function f is p-integrable if the function | f | ^p is integrable. 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 variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. The set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L² space.

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.Template:Mathanalysis-stub it:Funzione a quadrato sommabile pl:Funkcja całkowalna