Question: Let X; Y;X1;X2; : : : be real random variables on an event space . ;F/. Prove the following statements. (a) .X; Y / W

Let X; Y;X1;X2; : : : be real random variables on an event space . ;F/. Prove the following statements.

(a) .X; Y / W ! R2 is a random variable.

(b) X C Y and XY are random variables.

(c) supn2N Xn and lim supn!1 Xn are random variables (taking values in NR

).

(d) ¹X D Y º 2 F, ¹limn!1 Xn existsº 2 F, ¹X D limn!1 Xnº 2 F.

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