Dominated convergence theorem

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Lebesgue's Dominated Convergence Theorem. Let {fn} be a sequence of complex-valued measurable functions on a measure space (S, Σ, μ). Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that for all numbers n in the index set of the sequence and all points x ∈ S.Then f is integrable and