An example that the supremum of a family of measurable functions need not to be measurable.
Solution 1:
Let $E=[0,1]$, let $\Gamma\subset E$ be the Vitali set (which is uncountable), and for each $\gamma\in\Gamma$, let $$f_{\gamma}(t)=\mathbf{1}_{\{\gamma\}}(t)=\begin{cases}1 &\text{if }t=\gamma,\\ 0 &\text{if }t\neq \gamma. \end{cases}$$
Then $\sup_{\gamma}f_\gamma=\mathbf{1}_{\Gamma}$, which is not a measurable function because $\Gamma$ is not a measurable set.