Monotone convergence theorem

The monotone convergence theorem (MCT) is the foundational limit theorem of Lebesgue integration, establishing that for a sequence of non-negative measurable functions that increases pointwise to a limit, the integral of the limit equals the limit of the integrals. Unlike the dominated convergence theorem, which requires an external dominating function, the MCT relies solely on monotonicity and non-negativity — conditions that are often easier to verify in practice. The theorem is the engine by which Lebesgue integrals are constructed from simple functions: every measurable function is the limit of an increasing sequence of simple functions, and the MCT guarantees that the integral of the limit is the limit of the integrals of the approximations.

This construction is not merely a proof technique; it is the defining feature of the Lebesgue approach. Where the Riemann integral approximates from below and above simultaneously, requiring the two approximations to squeeze together, the Lebesgue integral approximates from below alone and uses the MCT to guarantee that the limit is correct. This asymmetry — lower bounds suffice, upper bounds are unnecessary — reflects a deep fact about measure spaces: non-negativity plus monotonicity creates a structure so rigid that convergence is automatic. The MCT is the mathematical expression of this rigidity.

The monotone convergence theorem reveals that in measure theory, order is stronger than topology. A monotone sequence converges in the integral sense even when it fails to converge uniformly, pointwise, or in measure. This is not a quirk of Lebesgue integration but a symptom of a broader principle: in ordered structures, monotonicity often implies convergence where topology cannot guarantee it. The same principle appears in the Knaster-Tarski theorem, in fixed-point theorems for ordered sets, and in the convergence of iterative methods in numerical analysis. The MCT is the measure-theoretic face of a much older idea: that direction implies destination.