40 1. Measure theory functions will be allowed to occasionally become infinite. However, as we will see, this can only happen on a set of Lebesgue measure zero.) To define the unsigned Lebesgue integral, we now turn to another more basic notion of integration, namely the b a f(x) dx of a Riemann integrable function f : [a, b] R. Recall from Section 1.1 that this integral is equal to the lower Darboux integral b a f(x) = b a f(x) dx := sup g≤f g piecewise constant p.c. b a g(x) dx. (It is also equal to the upper Darboux integral but much as the theory of Lebesgue measure is easiest to define by relying solely on outer measure and not on inner measure, the theory of the unsigned Lebesgue integral is easiest to define by relying solely on lower integrals rather than upper ones the upper integral is somewhat problematic when dealing with “improper” integrals of functions that are unbounded or are supported on sets of infinite measure.) Compare this formula also with (1.7). The integral p.c. b a g(x) dx is a piecewise constant integral, formed by breaking up the piecewise con- stant functions g, h into finite linear combinations of indicator functions 1I of intervals I, and then measuring the length of each interval. It turns out that virtually the same definition allows us to define a lower Lebesgue integral Rd f(x) dx of any unsigned function f : Rd [0, +∞], simply by replacing intervals with the more general class of Lebesgue mea- surable sets (and thus replacing piecewise constant functions with the more general class of simple functions). If the function is Lebesgue measurable (a concept that we will define presently), then we refer to the lower Lebesgue integral simply as the Lebesgue integral. As we shall see, it obeys all the basic properties one expects of an integral, such as monotonicity and additivity in subsequent notes we will also see that it behaves quite well with respect to limits, as we shall see by establishing the two basic convergence theorems of the unsigned Lebesgue integral, namely Fatou’s lemma (Corollary 1.4.46) and the monotone convergence theorem (Theorem 1.4.43). Once we have the theory of the unsigned Lebesgue integral, we will then be able to define the absolutely convergent Lebesgue integral, similarly to how the absolutely convergent infinite sum can be defined using the unsigned infinite sum. This integral also obeys all the basic properties one expects, such as linearity and compatibility with the more classical Riemann integral in subsequent notes we will see that it also obeys a fundamentally important convergence theorem, the dominated convergence theorem (Theorem 1.4.48). This convergence theorem makes the Lebesgue integral (and its abstract generalisations to other measure spaces than Rd) particularly suitable for
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