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Exercise 7 (#1.21). Let 12 = {wi : i = 1, 2,...} be a countable set, F be all subsets of 12, and v be the counting measure on 12 (i.e., v(A) = the number of elements in A for any ACN). For any Borel function f, the Chapter 1. Probability Theory integral of f w.r.t. v (if it exists) is | fdv = swi). i=1 Note. The definition of integration and properties of integration can be found in Shao (2003, $1.2). This type of exercise is much easier to solve if we first consider nonnegative functions (or simple nonnegative functions) and then general functions by using f+ and f-. See also the next exercise for another example. (#1.12). Let v and I be two measures on a o-field F on 12 such that v(A) = \(A) for any A C, where C C F is a collection having the property that if A and B are in C, then so is An B. Assume that there are Ai e C, i = 1, 2, ..., such that UA; N and v(Ai)