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. 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 if both f ⁺ and f ^- are measurable functions with finite Lebesgue integral. Let

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

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. 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 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.

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