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**emakarov** Let the probability space be [0, 1] with the standard measure. The idea is to have $\displaystyle X_n = n$ on an interval (or set) of measure 1/n and 0 everywhere else. Moreover, the support of $\displaystyle X_n$ (i.e., $\displaystyle \{\omega\in[0,1]: X_n(\omega)\ne0\}$) should shift visiting every point over and over again so that every $\displaystyle \omega$ is mapped to zero and non-zero arbitrarily far in the sequence. Then $\displaystyle X_n$ converges to 0 in probability because the measure of the support tends to zero. On the other hand, $\displaystyle X_n(\omega)$ does not converge for any $\displaystyle \omega$, so there is no almost sure convergence. Similarly, the measure of each $\displaystyle X_n$ is 1, so there is no convergence to 0 in $\displaystyle L^p$.

Can you give a precise definition of such $\displaystyle X_n$?