Pintegrable

Pintegrable, more commonly written as p-integrable, is a term used in measure theory and probability to describe functions whose p-th power is integrable. Let (X, Σ, μ) be a measure space. A measurable function f: X → R is p-integrable if ∫_X |f(x)|^p dμ(x) < ∞ for some p > 0. The collection of such functions forms the L^p space, denoted L^p(μ), and is equipped with the norm ∥f∥_p = (∫_X |f|^p dμ)^{1/p}.

For p ≥ 1, L^p(μ) is a Banach space. When 1 < p < ∞, L^p is also reflexive, and its dual space is L^q, where 1/p + 1/q = 1. The case p = ∞ is treated separately: L^∞ consists of essentially bounded functions with the norm ∥f∥_∞ = ess sup |f|, and is not defined by the p-th power integrability condition.

In probability theory, a random variable X is p-integrable if E|X|^p < ∞, making it an element of L^p on the underlying probability space. The most common instances are p = 1 (integrability in the expectation sense) and p = 2 (square integrability), the latter underpinning many techniques in statistics and signal processing.

If the underlying measure space has finite measure, higher p values imply stronger integrability relations: for p > q ≥ 1, L^p(μ) ⊂ L^q(μ). Other variants include L^p_loc, the space of functions that are p-integrable on every compact subset.

Pintegrable spaces thus provide a framework for quantifying function size via moments of order p and are central to analysis, probability, and applied mathematics.