Integrable function
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In mathematics, the term integrable function refers to a function whose integral may be calculated. Unless qualified, 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 measurable space X with sigma-algebra σ and measure μ, 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 L p spaces are one of the main objects of study of functional analysis.