Are Banach limits of measurable functions measurable?

Question

Let $(X, \mathcal{A}, \mu)$ be a probability space, and let $(f_k)_{k = 1}^\infty$ a sequence in $L^\infty$ such that $\|f_k\|_\infty \leq 1$. Let $L \in \left(\ell^\infty\right)^*$ be a Banach limit, and define a function $f : x \mapsto L \left( f_k(x) \right)_{k = 1}^\infty$. Is $f$ necessarily measurable? I frankly have no idea how to approach this question.

Answer 1

The limits of Borel measurable functions are Borel measurable: How to prove limit of measurable functions is measurable, but if you want it for any measure, you need the continuum hypothesis, Martin’s hypothesis, or other such additional axiom on top of ZFC. Seeing as you need more axioms to prove your statement, it isn’t necessarily the case that the Banach limits of measurable limits is measurable.

The medial limit I spoke of can be read about here: How to prove limit of measurable functions is measurable