Course in Probability Theory

3 .1 GENERAL DEFINITIONS 1 39by (2) . Now the collection of sets A in ?i2 for which (X, Y) - '(A) E T formsa B .F. by the analogue of Theorem 3 .1 .1 . It follows from what has just beenshown that this B .F . contains %~o hence it must also contain A2 . Hence eachset in /3 2 belongs to the collection, as was to be proved .Here are some important special cases of Theorems 3 .1 .4 and 3 .1 .5 .Throughout the book we shall use the notation for numbers as well as functions:(6) x v y= max(x, y), x A y= min(x, y) .Corollary . If X is an r .v . and f is a continuous function on R 1 , then f (X)is an r .v . ; in particular X r for positive integer r, 1XIr for positive real r, e-U,e"x for real A and t, are all r .v .'s (the last being complex-valued) . If X and Yare r, v .' s, thenXvY, XAY, X+Y, X-Y, X •Y ,X/Yare r .v .'s, the last provided Y does not vanish .Generalization to a finite number of r .v.'s is immediate . Passing to aninfinite sequence, let us state the following theorem, although its analogue inreal function theory should be well known to the reader .Theorem 3 .1 .6 . If {X j , j > 1} is a sequence of r .v .'s, theninf X j , sup X j, lim inf X j , lim sup XjJ J J jare r .v .'s, not necessarily finite-valued with probability one though everywheredefined, andlim X jj -+00is an r .v . on the set A on which there is either convergence or divergence to+0O .PROOF . To see, for example, that sup j X j is an r .v ., we need only observethe relationvdx E Jai' {supX j < x} = n{X j < x}JJand use Theorem 3 .1 .2 . Sincelim sup X j = inf (sup X j ),I n j>n