Fatou's lemma states that if 
is a sequence of nonnegative
measurable functions, then
 |
(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.](/images/equations/FatousLemma/NumberedEquation2.svg) |
(2)
|
Fatou's Lemma
See also
Almost Everywhere Convergence, Measure Theory, Pointwise ConvergenceExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Fatou's LemmaCite this as:
Weisstein, Eric W. "Fatou's Lemma." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FatousLemma.html