If a sequence of measurable functions converges pointwise and is dominated in absolute value by a single integrable function, then the limit of the integrals equals the integral of the limit. Proved by Henri Lebesgue, it is one of the central convergence theorems of measure theory.
Facts
StatementLebesgue's dominated convergence theorem gives a mild sufficient condition, that a sequence of functions is bounded in absolute value by a single integrable function, under which the limit of a sequence of integrals equals the integral of the limit. 1 Classification
Statement Form Connections
Sources
1. Dominated Convergence Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentenceQuote, Lead section, opening sentence
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged.
Lebesgue's Dominated Convergence Theorem (MathWorld)
Theorem statementQuote, Theorem statement
if {f_n} is a sequence of measurable functions, f_n converges pointwise to f almost everywhere as n→∞, and |f_n|≤g for all n, where g is Lebesgue integrable, then f is Lebesgue integrable, and ∫fdμ=lim_(n→∞)∫f_ndμ.
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