Let $S$ be a closed and bounded convex (not compact in general) subset of a Banach space, let $\{f_i\}_{i\in\mathcal I}$ be a (possibly uncountable) collection of affine continuous functions such that $\forall i\in\mathcal I, ~f_i(S)=[0,1]$. I want to know if \begin{align*} A\triangleq S\cap \bigcup_{i\in\mathcal I} f_i^{-1}(\{1\})=S\cap\bigcap_{n\in\mathbb N} \bigcup_{i\in \mathcal I} f_i^{-1}\left(\left] 1-\frac{1}{n},\infty \right[\right)\triangleq B \end{align*}
This is usefull in my research for knowing if $A$ is measurable, indeed $B$ is clearly measurable since $S$ is closed, the intersection is countable and the union of open sets is open.
It is clear that $A\subseteq B$. Let now $x\in B$, for a fixed $n$, let $\mathcal J_n=\left\{ i\in\mathcal I : 1-\frac{1}{n}<f_i(x)\right\}$, then for $n<m$, $\mathcal J_m\subseteq \mathcal J_n$ and since $x\in B$, $\bigcap_{n\in\mathbb N} \mathcal J_n\neq \emptyset$ (I don't feel comfortable with that step), now it is clear that $\bigcap_{n\in\mathbb N}\mathcal J_n = \{ i\in \mathcal I : 1\leq f_i(x) \}$, take any element $j$ of that set, you get $x\in S\cap f_j^{-1}(\{1\})\subseteq A$. Is that correct or close to be correct ?