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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
:$$
egin{array}{rl}
& f^+ = max (f,0) \
mbox{and} & f^- = max(-f,0)
end{array}

be the "positive" and "negative" part of ''f''. If ''f'' is integrable, then its integral is defined as
:$$
int f = mu(f^+ ) - mu(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'. 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.

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Square-integrable

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.