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 ?