58 1. Measure theory one works in the quotient space of L1(Rd) by the equivalence relation of almost everywhere agreement, which by abuse of notation is also denoted L1(Rd)), then one recovers a genuine metric. (Later on, we will establish the important fact that this metric makes the (quotient space) L1(Rd) a complete metric space, a fact known as the L1 Riesz-Fischer theorem this completeness is one of the main reasons we spend so much effort setting up Lebesgue integration theory in the first place.) The linearity properties of the unsigned integral induce analogous lin- earity properties of the absolutely convergent Lebesgue integral: Exercise 1.3.19 (Integration is linear). Show that the integration f Rd f(x) dx is a (complex) linear operation from L1(Rd) to C. In other words, show that Rd f(x) + g(x) dx = Rd f(x) dx + Rd g(x) dx and Rd cf(x) dx = c Rd f(x) dx for all absolutely integrable f, g : Rd C and complex numbers c. Also, establish the identity Rd f(x) dx = Rd f(x) dx, which makes integration not just a linear operation, but a *-linear operation. Exercise 1.3.20. Show that Exercises 1.3.15, 1.3.16, and 1.3.17 also hold for complex-valued, absolutely integrable functions rather than for unsigned measurable functions. Exercise 1.3.21 (Absolute summability is a special case of absolute inte- grability). Let (cn)n∈Z be a doubly infinite sequence of complex numbers, and let f : R C be the function f(x) := n∈Z cn1[n,n+1)(x) = c x where x is the greatest integer less than x. Show that f is absolutely integrable if and only if the series n∈Z cn is absolutely convergent, in which case one has R f(x) dx = n∈Z cn. We can localise the absolutely convergent integral to any measurable subset E of Rd. Indeed, if f : E C is a function, we say that f is measur- able (resp. absolutely integrable) if its extension ˜: f Rd C is measurable (resp. absolutely integrable), where ˜(x) f is defined to equal f(x) when x E and zero otherwise, and then we define E f(x) dx := Rd ˜(x) f dx. Thus,
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