Understanding supremum / infimum of a sequence of functions in context of sequences of measurable functions
It helps to be pedantic here:
- $\sup_n f_n(x)$ is not a function: it's a number. Specifically, it's the supremum of the set $\{f_1(x), f_2(x), \ldots\}$ which you should be familiar with from pre-measure theory real analysis.
- $\sup_n f_n$ is a function. Specifically, it's a function whose value at $x$ is $\sup_n f_n(x)$.
To be more clear, the author should have written "$\sup_n f_n$ is measurable".
You can define the other quantities analogously.
Hopefully this clears everything up for you.