Exercise 8 (#1.22). Let be a measure on a -field F on and f and g be Borel functions with respect to F. Show that (i) if f d exists and a R, then (af)d exists and is equal to a f d; (ii) if both f d and gd exist and f d + gd is well defined, then (f + g)d exists and is equal to f d + gd. Note. For integrals in calculus, properties such as (af)d =a f d and (f + g)d f d + gd are obvious. However, the proof of them are complicated for integrals defined on general measure spaces. As shown in this exercise, the proof often has to be broken into several steps: simple functions, nonnegative functions, and then general functions.