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Fatou's Lemma


If {f_n} is a sequence of nonnegative measurable functions, then

 intlim inf_(n->infty)f_ndmu<=lim inf_(n->infty)intf_ndmu.
(1)

An example of a sequence of functions for which the inequality becomes strict is given by

 f_n(x)={0   if x in [-n,n]; 1   otherwise.
(2)

See also

Almost Everywhere Convergence, Measure Theory, Pointwise Convergence

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References

Browder, A. Mathematical Analysis: An Introduction. New York: Springer-Verlag, 1996.Rudin, W. Ch. 1, Ex. 8 in Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, p. 23, 1987.Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.

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Fatou's Lemma

Cite this as:

Weisstein, Eric W. "Fatou's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FatousLemma.html

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