1.3. The Lebesgue integral 41 analysis, as well as allied fields that rely heavily on limits of functions, such as PDE, probability, and ergodic theory. Remark 1.3.1. This is not the only route to setting up the unsigned and absolutely convergent Lebesgue integrals. For instance, one can proceed with the unsigned integral but then making an auxiliary stop at integration of functions that are bounded and are supported on a set of finite mea- sure, before going to the absolutely convergent Lebesgue integral see e.g. [StSk2005]. Another approach (which will not be discussed here) is to take the metric completion of the Riemann integral with respect to the L1 metric. The Lebesgue integral and Lebesgue measure can be viewed as comple- tions of the Riemann integral and Jordan measure, respectively. This means three things. First, the Lebesgue theory extends the Riemann theory: every Jordan measurable set is Lebesgue measurable, and every Riemann inte- grable function is Lebesgue measurable, with the measures and integrals from the two theories being compatible. Conversely, the Lebesgue theory can be approximated by the Riemann theory as we saw in Section 1.2, ev- ery Lebesgue measurable set can be approximated (in various senses) by simpler sets, such as open sets or elementary sets, and in a similar fashion, Lebesgue measurable functions can be approximated by nicer functions, such as Riemann integrable or continuous functions. Finally, the Lebesgue the- ory is complete in various ways this is formalised in §1.3 of An epsilon of room, Vol. I, but the convergence theorems mentioned above already hint at this completeness. A related fact, known as Egorov’s theorem, asserts that a pointwise converging sequence of functions can be approximated as a (locally) uniformly converging sequence of functions. The facts listed here are manifestations of Littlewood’s three principles of real analysis (Section 1.3.5), which capture much of the essence of the Lebesgue theory. 1.3.1. Integration of simple functions. Much as the Riemann integral was set up by first using the integral for piecewise constant functions, the Lebesgue integral is set up using the integral for simple functions. Definition 1.3.2 (Simple function). A (complex-valued) simple function f : Rd C is a finite linear combination (1.8) f = c11E1 + . . . + ck1Ek of indicator functions 1Ei of Lebesgue measurable sets Ei Rd for i = 1,...,k, where k 0 is a natural number and c1,...,ck C are com- plex numbers. An unsigned simple function f : Rd [0, +∞], is defined similarly, but with the ci taking values in [0, +∞] rather than C. It is clear from construction that the space Simp(Rd) of complex-valued simple functions forms a complex vector space, and Simp(Rd) is also closed
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