There is no Borel function $f$, defined for all infinite sequences of 0's and 1's, such that for every sequence $X$ of 0-1 random variables that converges in probability to a constant $c$, we have $f(x) = c$ a.s.
D. Blackwell. "There are no Borel SPLIFs." Ann. Probab. 8 (6) 1189 - 1190, December, 1980. https://doi.org/10.1214/aop/1176994581