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Consider the triple (N, P(N), μ), where N denotes the set of natural numbers, P(N) is the power set of N, and for each subset A of N, μ(A) equals the number of elements in A. Show that μ is a measure (i.e., a nonnegative countably-additive set function, as in Defn. L.27) on P(N), which means that (N, P(N), μ) is a measure space. (This μ is an example of a counting measure.) What do the measurable functions look like? Have we studied these measurable functions before? What ones do we know how to “integrate” at our current stage in the theory of measure spaces? What do integrals with respect to counting measure look like?
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