I've been trying to learn a bit about analytic sets, and upper/lower semianalytic functions through the book, Stochastic optimal control, the discrete-time case.
In chapter 7, Lemma 7.30, it said that the supremum of a sequence of lower semianalytic functions $(f_n)$ is still lower semianalytic, as analytic sets are closed under countable intersections.
My questions is what if we have uncountably many lower semianalytic functions? It's quite unlikely the supremum is still lower semianalytic, as we don't have this nice closedness under countable intersections, but is there any sufficient condition for the supremum to be lower semianalytic?
Apologies if the question is elementry or unclear.