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On Wikipedia's entry of Convergence of Random Variables, it states that provided a probability space is complete:

If $X_n\ \xrightarrow{p}\ X$ and $X_n\ \xrightarrow{p}\ Y$, then $X=Y$ (almost surely).

However, I thought that this normally holds for the Borel measureable set instead of the Lebesgue (complete) measure space. Is the entry wrong?

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By almost surely it is meant almost everywhere $P$. The entry is right.

For a short proof of this fact, recall that every sequence converging in probability has a subsequence converging almost surely to it's limit. Applying this fact twice in a row we get $X_n\rightarrow X$ a.s. and $X_n\rightarrow Y$ a.s. along the same subsequence. In particular $X=Y$ a.s.

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    What I don't get is why they mention the probability space being complete. Do the results NOT hold if the space is not complete (ie, Borel measure)? – 2017-01-15
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    The assumption of completeness (i.e., every subset of a null-set is measurable) facilitates proofs since we don't have to worry about "almost sure" sets becoming non-measurable and hence not being able to speak about the probability of such event. – 2017-01-15