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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

and

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.

The L ^p spaces are one of the main objects of study of functional analysis.