 # 6. (a) Let 22 = Sh2 = the set of positive integers, F1 = F2 = all s...

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6. (a) Let 22 = Sh2 = the set of positive integers, F1 = F2 = all sub- sets, ui = 2 = counting measure, f (n, n) = n, f (n, n + 1) = -n, n = 1,2, f(i, j) = 0 if j # i or i + 1. Show that fair f du2 du = 0, for f du du2 = oo. (Fubini's theorem fails since the integral with respect to ui x 2 does not exist.) (b) Let St, = S2 = R. F1 = F2 = 38 (R), ui = Lebesgue measure, 12 = counting measure. Let A = {(w1, 02): wi = w2 € F1 x F2. Show that 21 12 IA = 1. (w) = 00, a but 22 2, Is du du2 = to R2 0. [Fubini's theorem fails since 12 is not o-finite; the product measure theorem fails also since for and ui do not agree on F1 x F2. 22

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