Hoi, I want to show that is measurable. Here is the closed ball with center x and radius r in some banach space X=(S,d), and is a "finite" borel-measure.

In the exercise we had to prove for some function , point-wise convergence (I shall not define f here since I believe only its properties are relevant). and also for any sequence we have

( is indicated as the indicator-function ;p)

I'm asked to prove the measurability using these results (not even sure if all the results are necessary). I'm somewhat stuck here. The only thing I can think of is writing the following:

and using the fact we may interchange limit and integral. I don't quite see yet how I might prove measurability...

Am i missing some heavy machinery here? Also, does the measurability of depend on the measurability of (i'm thinking of dominated convergence)? Do we need to know first whether is measurable?