42 1. Measure theory under pointwise product f, g fg and complex conjugation f f. In short, Simp(Rd) is a commutative ∗-algebra. Meanwhile, the space Simp+(Rd) of unsigned simple functions is a [0, +∞]-module it is closed under addition, and under scalar multiplication by elements in [0, +∞]. In this definition, we did not require the E1,...,Ek to be disjoint. How- ever, it is easy enough to arrange this, basically by exploiting Venn diagrams (or, to use fancier language, finite Boolean algebras). Indeed, any k subsets E1,...,Ek of Rd partition Rd into 2k disjoint sets, each of which is an inter- section of Ei or the complement Rd\Ei for i = 1,...,k (and in particular, is measurable). The (complex or unsigned) simple function is constant on each of these sets, and so can easily be decomposed as a linear combination of the indicator function of these sets. One easy consequence of this is that if f is a complex-valued simple function, then its absolute value |f|: x |f(x)| is an unsigned simple function. It is geometrically intuitive that we should define the integral Rd 1E(x)dx of an indicator function of a measurable set E to equal m(E): Rd 1E(x) dx = m(E). Using this and applying the laws of integration formally, we are led to pro- pose the following definition for the integral of an unsigned simple function: Definition 1.3.3 (Integral of a unsigned simple function). If f = c11E1 + . . . + ck1Ek is an unsigned simple function, the integral Simp Rd f(x) dx is defined by the formula Simp Rd f(x) dx := c1m(E1) + . . . + ckm(Ek), thus Simp Rd f(x) dx will take values in [0, +∞]. However, one has to actually check that this definition is well defined, in the sense that different representations f = c11E1 + . . . + ck1Ek = c11E1 + . . . + ck 1E k of a function as a finite unsigned combination of indicator functions of mea- surable sets will give the same value for the integral Simp Rd f(x) dx. This is the purpose of the following lemma: Lemma 1.3.4 (Well-definedness of simple integral). Let k, k 0 be natural numbers, c1,...,ck,c1,...,ck [0, +∞], and let E1,...,Ek,E1,...,Ek Rd be Lebesgue measurable sets such that the identity (1.9) c11E1 + . . . + ck1Ek = c11E1 + . . . + ck 1E k
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