This is a problem from the book Measure, Integration and Real Analysis by Axler.
Give an example of measurable space $(X,S)$ and a family ${f_t}$ $t$ belongs to $\mathbb{R}$ such that each $f_t$ is $S$ measurable function from $X$ to $[0,1]$ but the function $f(x) = \sup\{f_t(x)\}$ is not $S$ measurable.
If it would have been sequence of functions then $f$ can be shown to be $S$ measurable.
Best Answer
Let $(X, S)$ be the space of Lebesgue-measurable sets. Let $N$ be a non-Lebesgue measurable set. Then,
$$1_N = \sup_{x \in N} 1_{\{x\}}$$
is non-measurable, where $1_A$ is the indicator function for $A \subset X$.