Fatou's Lemma: Difference between revisions
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== | ==Statement== | ||
Suppose <math>\{f_n\}</math> is a sequence of non-negative measurable functions, <math> f_n: X \to [0,+\infty]</math>. | Suppose <math>\{f_n\}</math> is a sequence of non-negative measurable functions, <math> f_n: X \to [0,+\infty]</math>. | ||
Then: | Then: | ||
<math> \int \liminf_{n\rightarrow +\infty} f_n \leq \liminf_{n\rightarrow +\infty}\int f_n </math>. <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.2 </ref> | <math> \int \liminf_{n\rightarrow +\infty} f_n \leq \liminf_{n\rightarrow +\infty}\int f_n </math>. <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.2 </ref> | ||
==Proof== | |||
![{\displaystyle f_{n}:X\to [0,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c40303a8eaaeaa98d29ac2ac62dc40c14dd2f35)
