TITLE: Convergence to function that is not measurable
QUESTION [7 upvotes]: Suppose ($X,\cal M, \mu$) is not complete. Show that there is a sequence {$f_n$} of measurable functions on $X$ that converges pointwise a.e. on $X$ to a function $f$ that is not measurable.

REPLY [8 votes]: Pick a non-measurable subset $E$ of a null set, and let $f_n=0$ for all $n$. Then $f_n\to\chi_E$ a.e.