Weak convergence

Weak convergence is convergence defined by requiring a sequence of objects to converge when evaluated against every member of a specified class of test objects. The test class determines which features the convergence detects.

Convergence of measures: A notion of convergence for measures defined by how their integrals against test functions behave. Weak convergence of measures is a central instance of convergence defined through integrals.
Weak topology: The coarsest topology making a specified family of functionals continuous. It turns convergence under all chosen tests into ordinary topological convergence.
Strong convergence: Convergence measured by a norm or another strong notion of distance. Norm convergence implies weak convergence, but weak convergence need not control norms.
Weak convergence of probability measures: Convergence of probability measures against bounded continuous test functions. This standard form captures convergence in distribution of random quantities.
Uniform integrability: A condition preventing integrable functions from concentrating too much mass on sets of small measure. It helps ensure weak convergence of functions also controls convergence of integrals.
Test function: A function used to probe a mathematical object, often through integration or pairing. Test functions specify which features a weak limit must preserve.
Continuous linear functional: A linear map to the scalar field that is continuous in the given topology. These functionals are the probes defining weak convergence in normed spaces.
Pointwise convergence: Convergence of functions at each individual point of their domain. Pointwise tests and integral tests impose different conditions on sequences of functions.
Convergence in distribution: Convergence of random variables' probability laws, defined through bounded continuous test functions. It is the probabilistic use of weak convergence for laws.
Fatou's lemma: A measure-theoretic inequality placing the integral of a pointwise lower limit below a lower limit of integrals. Related lower-semicontinuity principles constrain quantities along weakly convergent sequences.

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