Dominated convergence theorem

The dominated convergence theorem states that pointwise convergence of measurable functions bounded in absolute value by an integrable function allows their integrals to converge to the integral of the limit.

Lebesgue integral: An integral defined by measuring the sizes of level sets of measurable functions. The theorem concerns convergence of Lebesgue integrals, whose measure-based definition supports its proof.
Measure space: A set equipped with a sigma-algebra and a measure assigning sizes to its measurable subsets. The theorem is formulated on measure spaces, not only on intervals with ordinary length.
Differentiation under the integral sign: A method for computing derivatives of parameter-dependent integrals by differentiating their integrands. Dominated convergence can justify limits of difference quotients needed for this method.
Monotone convergence theorem: A theorem stating that integrals commute with increasing limits of nonnegative measurable functions. It needs monotonicity instead of an integrable dominating function and allows infinite integrals.
Pointwise convergence: A sequence of functions converges pointwise when its values converge at each point of the domain. This is the convergence assumption the theorem combines with an integrable bound.
Measurable function: A function whose inverse images of measurable sets are measurable. Measurability makes the functions and their limit eligible for Lebesgue integration.
Fourier transform: An integral transform that represents a function through its frequency components. Convergence of Fourier integrands often uses domination to pass limits through the transform.
Uniform convergence: A sequence of functions converges uniformly when one error bound works across the entire domain. Uniform convergence can exchange limits and integrals on finite-measure spaces through a different control.
Almost-everywhere convergence: A sequence converges almost everywhere when it converges outside a set of measure zero. The theorem remains valid when pointwise convergence fails only on a null set.
Null set: A measurable set with measure zero. Exceptional failures of convergence on null sets do not affect the theorem's integral conclusion.

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