Let $(f_t)_{t \in \mathbb{R}}$ be a family of measurable functions on a measurable set $E$. Suppose that $\lim_{t \to 0} f_t(x) $ exists for all $x \in E$. Define $f(x) = \lim_{t \to 0} f_t(x)$.
Is f a measurable function?
My immediate response is to say “yes”. However, I have only seen results on taking limits $n \to \infty$ of countable sequences of measurable functions. It is not clear to me if the same results should stand here (e.g. that the limit of a sequence of measurable functions is also measurable), since I know uncountability has a bad habit of breaking things!
If the result is true, how can I extend the normal results about sequences of measurable functions to this case? If not, how do I find a counter example?