{ft},t varies from 0 to infinity, is a family of Lebesgue measurable functions that converges pointwise to f. Is f necessarily Lebesgue measurable?
Yes. The usual way to prove this is to show that $\displaystyle \limsup_{n\to\infty}f_n$ (or $\displaystyle \liminf_{n\to\infty}f_n$) is measurable, for any sequence of measurable functions. It follows that if the sequence has a pointwise limit then that must be measurable.