Let X be an infinite set and A be the algebra consisting of the finite and...

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Let X be an infinite set and A be the algebra consisting of the finite and co-finite subsets of X (cf. Prob.3). Define a set function u on A by setting for every ΑΕΑ: [0 if A is finite H(A) = { 1 if A is co-finite. (a) Show that u is additive. (b) Show that when X is countably infinite, u is not additive. (c) Show that when X is countably infinite, then X is the limit of an increasing sequence {Anne N} in A with u(An) = 0 for every n E N, but μ(X) = 1. (d) Show that when X is uncountably, the u is countably additive. Let X be an infinite set and A be the algebra consisting of the finite and co-finite subsets of X (cf. Prob.3). Define a set function u on A by setting for every ΑΕΑ: [0 if A is finite H(A) = { 1 if A is co-finite. (a) Show that u is additive. (b) Show that when X is countably infinite, u is not additive. (c) Show that when X is countably infinite, then X is the limit of an increasing sequence {Anne N} in A with u(An) = 0 for every n E N, but μ(X) = 1. (d) Show that when X is uncountably, the u is countably additive.

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