If X is a simple r v (not necessarily nonnegative), defined on the (product) space ((1 ( (2, A1 ( A2, ( 1 ( 2) with 1 and 2 finite, for which ( Xd ( exists, then show directly that ((( X ((1, (2) d 1 2 ((( X ((1, (2) d 2 d 1 ((( X ((1, (2) d( ( X d( ...

Let X n i1 x i I Ei where E 1 E n is a partition of 1 2 Then Xd n i1 x i E i and its existence means ...

If X is a simple r.v. (not necessarily nonnegative), defined on the (product) space ((1 ( (2,

Question:

If X is a simple r.v. (not necessarily nonnegative), defined on the (product) space ((1 ( (2, A1 ( A2, ( = μ1 ( μ2) with μ1 and μ2 σ-finite, for which ( Xd ( exists, then show directly that ((( X ((1, (2) d μ1 μ2 = ((( X ((1, (2) d μ2 d μ1 = ((( X ((1, (2) d( = ( X d(.