Consider the ( Iuml finite) measure space ((, A, Icirc frac14 ), and let Ai, i 1, 2, be a (measurable) partition of ( For each i, define the measure Icirc frac14 i by Icirc frac14 i (A) Icirc frac14 (A ( Ai) Then, if X is a r v defined on ((, A, Icirc frac14 ) for which the integral ( Xd Icirc frac14 exists, show that the integrals ( Xd Icirc frac14 , i ( 1, also exist and EIS Xd Xd Li 1 ntpx

Question: Consider the (Ï-finite) measure space ((, A, Î¼), and let {Ai, i = 1, 2,....} be a (measurable) partition of (. For each i, define

Consider the (Ïƒ-finite) measure space ((, A, Î¼), and let {Ai, i = 1, 2,....} be a (measurable) partition of (. For each i, define the measure Î¼i by: Î¼i (A) = Î¼ (A ( Ai). Then, if X is a r.v. defined on ((, A, Î¼) for which the integral ( Xd Î¼ exists, show that the integrals ( Xd Î¼, i ( 1, also exist and

EIS Xd = Xd . Li=1 !ntpx

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