Question: (10) Let X = Y = (0,1). Let A = B be the family of Borel subsets of this space, let y = m be

(10) Let X = Y = (0,1). Let A = B be the family of Borel subsets of this space, let y = m be one-dimensional Lebesgue measure, and let v be the counting measure on (Y,B) so that {{z}) = 1 for each x (0,1). Let G = {(x,x?): (0.1]} X XY. (a) Show that G and Xo are measurable wrt hxv. (b) Compute Sy (Sx Xc(2,y) du(x)) dv(y) and Sx Oy Xc (,y) dv(y)) du(x) and show that these integrals are not equal. (c) Explain why the result of part (b) does not contradict Tonelli's Theorem. (10) Let X = Y = (0,1). Let A = B be the family of Borel subsets of this space, let y = m be one-dimensional Lebesgue measure, and let v be the counting measure on (Y,B) so that {{z}) = 1 for each x (0,1). Let G = {(x,x?): (0.1]} X XY. (a) Show that G and Xo are measurable wrt hxv. (b) Compute Sy (Sx Xc(2,y) du(x)) dv(y) and Sx Oy Xc (,y) dv(y)) du(x) and show that these integrals are not equal. (c) Explain why the result of part (b) does not contradict Tonelli's Theorem

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