integrable

Integrable is a term used across mathematics to describe objects for which an integral can be meaningfully defined, or more broadly, that admit explicit solutions by integration. The precise meaning depends on context.

In real analysis, a function f on an interval is Riemann integrable if the limit of Riemann sums exists. Lebesgue integrability requires the integral of |f| to be finite, so f belongs to L1. Improper integrals extend these ideas to unbounded domains or to functions with unbounded behavior. A Riemann-integrable function may be discontinuous, but its set of discontinuities has measure zero.

In probability theory, a random variable X is integrable when E|X| < ∞; then the expectation E[X] exists. Integrability ensures that averages, moments, and certain convergence results are well defined, and it underpins many limit theorems and statistical methods.

In dynamical systems and physics, an integrable system is one that possesses as many independent conserved quantities in involution as degrees of freedom, allowing the equations of motion to be solved by quadrature. Liouville integrability implies the existence of action-angle variables; classical examples include the harmonic oscillator and certain celestial mechanics problems. Techniques associated with integrable systems include Lax pairs and spectral methods. By contrast, many generic systems are non-integrable and can exhibit chaotic behavior.

Thus, integrable captures the notion of solvability or well-behaved integration across topics, from function theory to dynamics, probability, and numerical analysis.