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?
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4Does this answer your question? supremum of uncountable families of Borel measurable functions – LL 3.14 Jul 19 '20 at 09:43
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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.
Kavi Rama Murthy
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