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Suppose that $\{f_c(x)\}$ for all $c\in [a, b]$ is a family of measurable functions defined on $\mathbb R$.

Suppose that for each $x$, $c\mapsto f_c(x)$ is continuous.

Show that the function $g(x)= \sup{\{f_c(x): c\in [a,b] \}}$ is measurable.

In my opinion only supreme of countable collection measurable functions is measurable. but here we have uncountable collection.so where do i use the continuity of $g(x)$?

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  • $\begingroup$ Everything you say is correct. But continuity implies that the uncountable supremum can be obtained by taking the supremum over a countable subfamily (how?) $\endgroup$ – PhoemueX Nov 7 '15 at 10:52
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Hint: Let $u\in \mathbb{R}$, $A=\{x; g(x)>u\}$, $\displaystyle B=\cup_{c\in \mathbb{Q}\cap[a,b]}\{x;f_c(x)>u\}$. Show that $A=B$.

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  • $\begingroup$ Very nice hint. $\endgroup$ – Ramiro Nov 7 '15 at 16:13

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