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MIT18_102s09_lec07

# MIT18_102s09_lec07 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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41 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 7. Thursday, Feb 26 So, what was it with my little melt-down? I went too cheap on the monotonicity theorem and so was under-powered for Fatou’s Lemma. In my defense, I was trying to modify things on-the-ﬂy to conform to how we are doing things here. I should also point out that at least one person in the audience made a comment which amounted to pointing out my error. So, here is something closer to what I should have said it is not far from what I did say of course. Proposition 12. [Montonicity again] If f j ∈ L 1 ( R ) is a monotone sequence, either f j ( x ) f j +1 ( x ) for all x R and all j or f j ( x ) f j +1 ( x ) for all x R and all j, and f j is bounded then (7.1) { x R ; lim f j ( x ) is finite } = R \ E j →∞ where E has measure zero and f = lim f j ( x ) a.e. is an element of L 1 ( R ) (7.2) j →∞ with lim f f j = 0 . j →∞ | | Moral of the story drop the assumption of positivity and replace it with the bound on the integral. In the approach through measure theory this is not necessary because one has the concept of a measureable, non-negative, function for which the integral ‘exists but is infinite’ we do not have this. Proof. Since we can change the sign of the f i (now) it suﬃces to assume that the f i are monotonically increasing. The sequence of integrals is therefore also montonic increasing and, being bounded, converges. Thus we can pass to a subsequence g i = f n i with the property that (7.3) | g j g j 1 | = g j g j 1 < 2 j j > 1 . This means that the series h 1 = g 1 , h j = g j g j 1 , j > 1 , is absolutely summable. So we know for the result last time that it converges a.e., that the limit, f, is integrable and that j (7.4) f = lim h k = lim g j = lim f j . j →∞ k =1 j →∞ n →∞ In fact, everywhere that the series h j ( x ) , which is to say the sequence g k ( x ) , j converges so does f n ( x ) , since the former is a subsequence of the latter which is monotonic. So we have (7.1) and the first part of (7.2). The second part, corresponding to convergence for the equivalence classes in L 1 ( R ) follows from monotonicity, since (7.5) | f f j | = f f j 0 as j → ∞ .
42 LECTURE NOTES FOR 18.102, SPRING 2009 Now, to Fatou’s Lemma. This really just takes the

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MIT18_102s09_lec07 - MIT OpenCourseWare http/ocw.mit.edu...

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