AssumeX, Y are random variables, defined on some probability space (2, F,P), so that E[|X|] +E[|Y...

 

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AssumeX, Y are random variables, defined on some probability space (2, F,P), so that E[|X|] +E[|Y |] < o. Assume E,G CF are o-algebras. A. always, E[Y |G] = E[Y\o(EUG)], P-a.s.; B. if E CG and o(Y) = G, then E[XY|G] = YE[X|G)], P-a.s.; %3D C. if E CG and o(Y) = G, then E[XY|E] = YE[X|E)], P-a.s.; D. if E CG and E # G, then E[X|E] < E[X]G], P-a.s.. Here P-a.s. indicates that, with probability one, an identity or (strict) inequality holds for the associated versions. AssumeX, Y are random variables, defined on some probability space (2, F,P), so that E[|X|] +E[|Y |] < o. Assume E,G CF are o-algebras. A. always, E[Y |G] = E[Y\o(EUG)], P-a.s.; B. if E CG and o(Y) = G, then E[XY|G] = YE[X|G)], P-a.s.; %3D C. if E CG and o(Y) = G, then E[XY|E] = YE[X|E)], P-a.s.; D. if E CG and E # G, then E[X|E] < E[X]G], P-a.s.. Here P-a.s. indicates that, with probability one, an identity or (strict) inequality holds for the associated versions.