We toss two dice with sample space Ω = {(i, j), 1 ≤ i, j ≤ 6} and the σ-algebra is generated by the events Ak = {(i,j) : max(i,j) = k} (k = 1, . . . , 6). Show that whereas X1(i, j) = max(i, j) is a random variable in the corresponding probability space, X2(i, j) = i + j is not a random variable.

I know that to be a random variable, X1 and X2 need to have a pre-image in the event space Sigma. X1 can take on values from 1 to 6 depending on the highest number in the (i,j) pair. X2 can take on values from 2 (1,1) to 12 (6,6).

So if i follow the rule, one element outcome of X1 and X2 should have an outcome which is an element of Sigma as a pre-image. I understand that each value of X1 (1 up to 6) is a possible outcome of Sigma. But it is less clear for X2, if I have a sum equals to 12, should I say it's not a random variable because there's not 12 in the Sigma even if the Sigma includes (6,6) ?