Real and complex analysis

ABSTRACT INTEGRATION 19The cancellation laws have to be treated with some care: a + b = a + cimplies b = c only when a < 00, and ab = ac implies b = c only when 0 < a < 00.Observe that the following useful proposition holds:If we combine this with Theorems 1.17 and 1.14, we see that sums and productsof measurable functions into [0, 00] are measurable.Integration of Positive FunctionsIn this section, IDl will be a a-algebra in a set X and p. will be a positive measureon IDl.1.23 Definition If s: X -4 [0, 00) is a measurable simple function, of the formnS = L exiXAi'i= 1(1)where ex 1 , ••. , ex n are the distinct values of s (compare Definition 1.16), and ifE E IDl, we define(2)The convention 0 . 00 = 0 is used here; it may happen that ex; = 0 for some iand that P.(Ai 11 E) = 00.Iff: X -4 [0, 00] is measurable, and E E IDl, we defineif dp. = sup is dp., (3)the supremum being taken over all simple measurable functions s such thatO:::;s:::;1The left member of (3) is called the Lebesgue integral of f over E, withrespect to the measure p.. It is a number in [0, 00].Observe that we apparently have two definitions for IE f dp. iffis simple,namely, (2) and (3). However, these assign the same value to the integral,since f is, in this case, the largest of the functions s which occur on the rightof (3).1.24 The following propositions are immediate consequences of the definitions.The functions and sets occurring in them are assumed to be measurable:(a) If 0 :::;f:::; g, then IE f dp. :::; IE g dp..(b) If A c B andf:2:. 0, then IA f dp. :::; IB f dp..