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Is the limit of a $L^2$-convergent sequence of random variables unique up to a.e.? In other words, if $X$ and $Y$ are both limits, will $X=Y$ a.e.? If yes, is Ito integral, which is defined as $L^2$ limit of a sequence of Ito integrals of simple processes, defined only up to a.e.?

Conversely, if the sequence converges to a random variable $X$, and $Y$ is another random variable same as $X$ a.e., will $Y$ also be the limit of the $L^2$-convergent sequence?

Similar questions for a sequence of random variables that converges in probability.

Thanks in advance!

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