THE PRODUCT OF STRONG OPERATOR MEASURABLE FUNCTIONS 1099 (ii) The Bochner integral has certain extremely useful properties [9, Theorems 3.7.4, 3.7.9, 3.7.13, pp. a)Prove that the product of two measurable extended real valued function is measurable b)If f and g are measurable extended real-valued functions and α a fixed number, then f+g is measurable if we define f+g to be α whenever it is of the form ∞-∞ or -∞+∞ Thanks for your help This result, called Fubini’s theorem, is another one of the basic and most useful properties of the Lebesgue integral. Then (X,S) is a measurable space. Thus f is measurable if and only if {x| f(x) >α} ∈ A for every α∈ R. Corollary 4.1.1. We want to follow the idea of Riemann sums and introduce the idea of a Lebesgue sum of “rectangles” whose heights are determined by a function and whose base is determined by the measure of a set. This has occurred and has sometimes been explicitly pointed out in the work of the author and his coauthors on or related to the "Feynman integral" (for example, [4, p. 1325, … Measurable functions are functions that we can integrate with respect to measures in much the same way that continuous functions can be integrated \dx". We want to guarantee that the sets which arise when working with these functions are measurable. A subset E of X is said to be measurable if E ∈ S. In this chapter, we will consider functions from X to IR, where IR := IR∪{−∞}∪{+∞} is the set of extended real numbers. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001 [Hal] [Bor] E. Borel, "Leçons sur la theorie des fonctions" , Gauthier-Villars (1898) Zbl 29.0336.01 [Bou] N. Bourbaki, "Elements of mathematics. f2, the product of two measurable functions • f1 f2 where f2 is nowhere zero In particular, −f1 is also measurable. Proposition 3.1. Integration" , Addison-Wesley (1975) pp. Lebesgue Measurable Functions Section 3.1. Measurable Functions §1.

80-84], which often does make its use essential. A continuous function pulls back open sets to open sets, while a measurable function pulls back measurable sets to measurable sets. Sums, Products, and Compositions Note. Measurability Most of the theory of measurable functions and integration does not depend on the speci c features of the measure space on which the functions …

Recall that the Riemann integral of a continuous function fover a bounded interval is de ned as a limit of sums of lengths of subintervals times values of fon the subintervals. The integral of a measurable function on the product space may be evaluated as iterated integrals on the individual spaces provided that the function is positive or integrable (and the measure spaces are ˙- nite). For simplicity, we write ∞ for +∞. Measurable Functions Let X be a nonempty set, and let S be a σ-algebra of subsets of X. 3.1. Measurable functions in measure theory are analogous to continuous functions in topology. Measurable Functions Dung Le1 1 Deﬁnition It is necessary to determine the class of functions that will be considered for the Lebesgue integration.