My question is about the last two line of the proof of the part (1):
How did the author proceed from the second last line to the last line?
First I thought he just took the summation into the integral. However, to do so one have to show first that the integrand converge uniformly.
Then I thought the integral defined on the last line may be in the 'extended sense', that is, defined through a series which take the form similar to the second last line. But to do so, I think one must shows that $$\{\psi \circ g\} $$ is a partition of unity of the set $A$. I think this is doable but I didn't see it in the proof.