Integrable function

Given a measurable space X with sigma-algebra σ and measure μ, a real valued function f:X → R is integrable if both f ⁺ and f ^- are measurable functions with finite integral. Recall that

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

are 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.

The L ^p-spaces are one of the main objects of study of Functional Analysis.