# Measurabilty of supremum [duplicate]

Consider a measurable space $$(\Omega,\mathcal{F})$$.

Is the supremum of an uncountable family of Borel measurable functions of the type $$f:\Omega\to\mathbb{R}$$ a Borel measurable function?

Let $$E$$ be any nonmeasurable set and $$f_t(x)=1$$ if $$t=x$$ and $$0$$ otherwise. Then $$sup \{f_t(x):t \in E\}=\chi_E$$ which is not measurable but each $$f_t$$ is measurable.