Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems

MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez

We define the Lebesgue integral for non-negative functions via a limiting process involving simple functions. We then prove two fundamental results of Lebesgue integration: the Monotone Convergence Theorem and Fatou’s Lemma.

License: Creative Commons BY-NC-SA

[ktayeb]

In the proof of the theorem of the convergence monotone, there is the case where f(x)=0, which is easy to discuss

[adilsanaulla]

I think in the last mct, we also need assumption that f be nonnegative on E, because otherwise its integral over E is not defined since it is not in L+(E). It may be measurable and also nonnegative ae but I don’t see why it has to be nonnegative on all of E.

[hausdorffm]

1:00:10 Can we remove the assumption that f_n converges to f in the monotone convergence theorem?
The reason why I guess so is that now each f_n is positive and we consider it as a extended real valued function,
so it always conveges in the sence of extended valued function.

[nullspace]

Make the whole class an exercise and be done.