Monotone Convergence Theorem: Difference between revisions
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==Monotone Convergence Theorem== | ==Monotone Convergence Theorem== | ||
Suppose <math>\{f_n\}</math> is a sequence of non-negative measurable functions, <math> f_n: X \to [0,+\infty]</math> such that <math> f_{n-1} \leq f_{n} </math> for all <math>n</math>. | Suppose <math>\{f_n\}</math> is a sequence of non-negative measurable functions, <math> f_n: X \to [0,+\infty]</math> such that <math> f_{n-1} \leq f_{n} </math> for all <math>n</math>. | ||
Furthermore, <math> \lim_{n\to+\infty} f_n = f ( = \sup_n f_n) </math>. | Furthermore, <math> \lim_{n\to+\infty} f_n = f ( = \sup_n f_n) </math>. | ||
Then | Then | ||
:<math> \lim_{n\to+\infty} \int f_n = \int \lim_{n\to+\infty} f_n .</math><ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.2 </ref> | :<math> \lim_{n\to+\infty} \int f_n = \int \lim_{n\to+\infty} f_n .</math><ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.2 </ref> | ||
==Proof== | |||
==References== | |||
==References== | ==References== | ||
![{\displaystyle f_{n}:X\to [0,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c40303a8eaaeaa98d29ac2ac62dc40c14dd2f35)
