Questions about measurability of random variables and expectation

Consider a sample space Q consisting of countably many sample points (from a random ex periment). And set 8 as the collection of all subsets of (I, called events. Then, a probability measure on this measurable space (S2, 8) is nothing but the assignment of nonnegative prob ability masses P({:c}) for each sample point 3; E 9 such that 29,653, P({I}) : 1. Now consider any function f : S! > R (random variable). Show that, rst, this f is 8- measurable, and second, [9 deP = Z f(m)P({:v})- 236 Q Note that this is the mathematical expectation of f, usually denoted by Elf]