xo-o fd(X) = fd(XO) for each Xo E (a, bj. , 'L /':;f(Cj) < c. 9) x:S;Cj

By assuming that JL(0) = 0 and JL(A) = +00 for A =1= 0. Clearly, this measure is not a-finite. 2. Let R = N. On the a-algebra 9Jt(N) we define a measure JL by assuming JL(A) = n if a set A consists of n elements, and JL(A) = +00 if A is an infinite set. It is easy to verify that JL is a a-finite measure; for the sets An, we can take, for example, {1,2, ... ,n}. An important example of a a-finite measure is considered in Section 10. 1. Show that the condition Al <;;; A2 <;;; A3 C ... 2 can be omitted.

N}. An important example of a a-finite measure is considered in Section 10. 1. Show that the condition Al <;;; A2 <;;; A3 C ... 2 can be omitted. 2. Show that if a measure JL given on an algebra ~ of subsets of R is afinite, then R can be represented in the form of a union of countably many pairwise disjoint sets with finite measures. 3. Let R = N. Assume that JL(A) = 0 if A is a finite set, and JL(A) = +00 if A is infinite. Prove that JL is an additive function on 9Jt(N). Is JL a measure? 4. Let f : JR.