If ff ngis a sequence of measurable functions on X, then fx: limf n(x) existsgis a measurable set. That means pointwise convergence almost everywhere, i.e. PROBLEM 6. Then we say that f n converges to f (1) almost everywhere (a.e.) The following theorem gives a similar result for nonnegative valued measurable functions. 0 is measurable since fx: jf n(x) f m(x)j<1=kgis. measurable functions (if |f(x)| ≤ C for all x, we can get an increasing sequence of simple functions converging to it pointwise (even uniformly, in fact), all of which take values between 0 and C). The pointwise limit of a sequence of measurable functions : →. Theorem 2. if f n(x) → f(x) for all x … Fatou's Lemma: Let {f} be a sequence of nonnegative measurable functions on E. Then, ſliminf /, slimin ſa Proof: Let = inff. We characterize convergence in measure of a sequence (f n )n of measurable functions to a measurable function f by elements of c 0, which express the quality of convergence of (f n )n to f. • By taking differences of nonnegative bounded measurable functions, all bounded measurable functions (if f is bounded, so are f+ and f−). Prove that ff ng!fin measure on Eif and only if every subsequence of ff nghas in turn a further subsequence that converges to fpointwise a.e. Recall that the Riemann integral of a continuous function fover … Math 2210 Real Analysis Problem Set 3 Solutions I. Minevich (small corrections by R. Kenyon) November 27, 2009 p. 48 # 3. Suppose (X,F,µ) is a σ-finite measure space and let f be a measurable function defined on X. are measurable if the original sequence (f k) k, where k ∈ ℕ, consists of measurable functions. Convergence of sequences of functions Definition Let {f n} be a sequence of measurable functions defined on a measure space (X,µ), and let f be another measurable function. Let fbe any measurable function E. Show that there is a sequence of semisimple functions ff ngon Ethat converge to funiformly on E. The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well. Let fbe any measurable function E. Show that there is a sequence of semisimple functions ff ngon Ethat converge to funiformly on E. n is a sequence of measurable functions that converge to a real-valued function fa.e. Problem 13: A real valued measurable function is said to be semisimple provided it takes only a countable number of values. there exits M>0 such that jf n(x)j λ}, λ > 0, is non-increasing and right-continuous. Fix x2Eand de ne: f (x) = sup n 1 f n(x) f

The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.