$\{f_n\}_1^\infty$ are measurable,$\{f_n\}\rightarrow_{\mu.a.s} f$. Does $f$ is measurable in case $(X,\mathcal{F})$ isn't a complete space?

Question

$\{f_n\}_1^\infty$ are $(X,\mathcal{F})\rightarrow(\mathbb{R},\mathcal{B}_{})$ measurable, and $\lim_{n\rightarrow \infty} \{f_n\}= f$ $\mu.a.s$. Does $f$ is measurable in case $(X,\mathcal{F})$ is not a complete measurable space? If not, are there any additional conditions in which this might be true in incomplete measurable space?