I am working with a set $I$, defined as the closed interval $[0,1]$, and a set $X$, which consists of all bounded Borel measurable functions defined on $[0,1]$. The uniform metric on $X$ is defined by:$$d(f,g) := \sup_{x \in [0,1]} |f(x) - g(x)|,$$and we consider the associated Borel $\sigma$-algebra $\mathcal{B}$ on $X$.
I am interested in understanding the measurability of the map $\tau$ defined by:$$\tau: I \times X \to \mathbb{R}$$$$ (x,f) \mapsto f(x) $$Specifically, I would like to know whether this map $\tau$ is Borel measurable. This entails proving whether the set $\{(x,f) \in I \times X \mid f(x) \in (a,b)\}$ is in the $\sigma$-algebra generated by the product $\sigma$-algebra of $I$ and $X$.
From previous knowledge, I know that if $X$ is replaced with the collection of continuous functions $C[0,1]$, then the map $\tau$ is continuous. However, I am unsure how to proceed with the bounded Borel measurable functions in $X$. Could someone provide insights or guidance on how to approach this problem?