Beppo-Levi Theorem

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The Beppo-Levi theorem is a result in measure theory which gives a sufficient condition for interchanging an integral with an infinite series. The setting and result is essentially a particular case of the monotone convergence theorem, though one needs to be careful that all intermediary functions in the proof remain measurable so that monotone convergence may be applied.


Let be the underlying measure space and let be a sequence of measurable functions with for each . Then,


We know for any two non-negative measurable functions that

Iterating this formula inductively, we find for all that
In addition, we know that the sum of two nonnegative measurable functions is again nonnegative and measurable, and induction implies that each is again measurable and nonnegative.

The sequence of functions is monotonically nondecreasing since each is nonnegative. By the monotone convergence theorem, we thus deduce


1. Folland, Gerald. B; "Real Analysis: Modern Techniques and Their Applications." Wiley. 2007.