Lebesgue measure
In the previous chapter we discussed differentiation in several variable calculus. It is now only natural to consider the question of integration in several variable calculus.
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In the previous chapter we discussed differentiation in several variable calculus. It is now only natural to consider the question of integration in several variable calculus. The general question we wish to answer is this: given some subset Ω of Rn , and some real-valued function f : Ω → R, is it possible to integrate f on Ω to obtain some number Ω f ? (It is possible to consider other types of functions, such as complex-valued or vector-valued functions, but this turns out not to be too difficult once one knows how to integrate real-valued functions, since one can integrate a complex or vector valued function, by integrating each realvalued component of that function separately.) In one dimension we already have developed (in Chapter 11) the notion of a Riemann integral [a,b] f , which answers this question when Ω is an interval Ω = [a, b], and f is Riemann integrable. Exactly what Riemann integrability means is not important here, but let us just remark that every piecewise continuous function is Riemann integrable, and in particular every piecewise constant function is Riemann integrable. However, not all functions are Riemann integrable. It is possible to extend this notion of a Riemann integral to higher dimensions, but it requires quite a bit of effort and one can still only integrate “Riemann integrable” functions, which turn out to be a rather unsatisfactorily small class of functions. (For instance, the pointwise limit of Riemann integrable functions need not be Riemann integrable, and the same goes for an L2 limit, although we have already seen that uniform limits of Riemann integrable functions remain Riemann integrable.) Because of this, we must look beyond the Riemann integral to obtain a truly satisfactory notion of integration, one that can handle even very discontinuous functions. This leads to the notion of the Lebesgue integral, which we shall spend this chapter and the next constructing. The © Springer Science+Business Media Singapore 2016 and Hindustan Book Agency 2015 T. Tao, Analysis II, Texts and Readings in Mathematics 38, DOI 10.1007/978-981-10-1804-6_7
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163 Lebesgue integral can handle a very large class of functions, including all the Riemann integrable functions but also many others as well; in fact, it is safe to say that it can integrate virtually any function that one actually needs in mathematics, at least if one works on Euclidean spaces and everything is absolutely integrable. (If one assumes the axiom of choice, then there are still some pathological functions one can construct which cannot be integrated by the Lebesgue integral, but these functions will not come up in real-life applications.) Before we turn to the details, we begin with an informal discussion. In order to understand how to compute an integral Ω f , we must first understand a more basic and fundamental question: how does one compute the length/area/volume of Ω? To see why this question is connected to that of integration, observe that if one integrates the function 1 on the set Ω, then one should obtain the
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