Let ( = {0, 1, 2,...}, let A = {1, 3, 5,...}, and let C = {(, A, Ac, (}. Let μ be the counting measure on C, and let μ1, μ2 be defined on P(() by
μ1 (B) = the number of points of B, μ2 (B) = 2 μ1 (B).
Then show that
(i) C is a field.
(ii) μ is not σ-finite on C.
(iii) Both μ1 and μ2 are extensions of μ and are also σ-finite.
(iv) Determine the outer measure μ* by showing that μ* (B) = ( whenever B ( (.
(v) Show that the σ-field of μ*-measureable sets, A* say, is equal to P(().
(From this example, we conclude that if μ is not σ-finite on the field C, then there need not be a unique extension. Also, there may be σ-finite extensions, such as μ1 and μ2 here, when the original measure on C is not σ-finite.)