Monotone Convergence Theorem

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Consider the measure space and suppose is a sequence of non-negative measurable functions, such that for all . Furthermore, . Then



First we prove that .

Since for all , we have and further .

Sending on LHS gives us the result.

Then we only need to prove that . In this regard, denote (which is measurable as the limit of measurable functions) and consider a simple function so that .

Now, for and some , define the increasing sequence of measurable sets . Since , we have for . Moreover, we have since .

Recall that the set function for is a measure on . We now see that , where we have used the fact that for the last equality. Taking the supremum over simple functions such that and the supremum over yields the required inequality.


  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2