Clearly, the pointwise limit of a sequences of measurable functions is measurable: $$f_n\text{ measurable}\implies f\text{ measurable}\quad(f_n\to f\text{ pointwise})$$ (Especially, this holds true for uniform limit of a sequence.)
On the other hand this fails for nets in general, e.g.: $$f_{E\in\mathcal{B}([0,1]):E\subseteq V}:=\chi_E:\quad f_E\text{ measurable},f\text{ nonmeasurable}$$ with a Vitali set $V$ of the unit interval $[0,1]$.
But what about the uniform limit of a net: $$f_\lambda\text{ measurable}\implies f\text{ measurable}\quad(f_\lambda\to f\text{ uniformly})$$ This, especially, would imply that the bounded Borel functions form a Banach space under the supremum norm...
Choose $\{ f_{\alpha_{n}}\}_{n=1}^{\infty}$ such that $\sup |f_{\alpha_{n}}-f| < 1/n$.