Fatou's Lemma

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Suppose is a sequence of non-negative measurable functions, . Then:

. [1]


Define for all .

By definition, and , so by Monotone Convergence Theorem,


Furthermore, by definition we have , implying that .

Since exists, taking of both sides yields:



  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2
  2. Craig, Katy. MATH 201A Lecture 14. UC Santa Barbara, Fall 2020.