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I'm having a little trouble with a probability problem with three parts; I think I'm having trouble wrapping my head around just what's going on here. If anyone could give me a starting point, I'd appreciate it.

Here's the problem (Billingsley 5.1) (X a random variable)

a. Show that X is measurable w.r.t. the sigma field J iff sigma(X) is a subset of J. Show that X is a measurable w.r.t. sigma(Y) iff sigma(x) is a subset of sigma(Y)

b. Show that if J = {empty set, omega}, then X is measurable w.r.t. J iff X is constant.

c. Suppose that P(A) is 0 or 1 for every A in J. This holds, for example, if J is the tail field of an independent sequence, or if J consists of the countable and cocountable sets on the unit interval with Lebesgue measure. Show that if X is measurable w.r.t. J, then P[X=c] = 1 for some constant c.

Thanks for any and all help!

Best regards

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# Random Variable Measurability w.r.t. Sigma Fields

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