Let the probability space be [0, 1] with the standard measure. The idea is to have

on an interval (or set) of measure 1/n and 0 everywhere else. Moreover, the support of

(i.e.,

) should shift visiting every point over and over again so that every

is mapped to zero and non-zero arbitrarily far in the sequence. Then

converges to 0 in probability because the measure of the support tends to zero. On the other hand,

does not converge for any

, so there is no almost sure convergence. Similarly, the measure of each

is 1, so there is no convergence to 0 in

.

Can you give a precise definition of such

?