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Monotone Convergence Theorem

The monotone convergence theorem states that if {f_n} is a sequence of measurable functions, with 0<=f_n<=f_(n+1) for every n, then

intlim_(n->infty)f_ndmu=lim_(n->infty)intf_ndmu.

This measure-theoretic result is also called the Beppo Levi theorem. It is distinct from the monotone sequence theorem for bounded monotonic sequences.

See also

Fatou's Lemma, Lebesgue Integral, Lebesgue's Dominated Convergence Theorem, Measurable Function

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References

Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, 1988.

Referenced on Wolfram|Alpha

Monotone Convergence Theorem

Cite this as:

Weisstein, Eric W. "Monotone Convergence Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MonotoneConvergenceTheorem.html

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