Suppose that Ian } is a sequence of nonnegative numbers such that an -+ 0 and EnTh an = oo. Let f be a nonnegative measurable function on a measurable space (X, g”). Show that there are measurable sets {An } such that
co f = E an ‘An. n=i
HINT. First prove the following. For any positive real number f, there is a subset I of positive integers such that f = Env an. Find a good algorithm for obtaining J in terms of f and use it to define the sets An.