# Convergence of random variables, which properties hold only for a complete measurable space?

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?

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.