Is there any counter example for for $F_n \downarrow F$ then $\int F_n \, d\mu \downarrow \int F \, d\mu$?
I came up with the one below but $F_n$ does not go down to $0$ monotonically. I need something that goes monotonically.
$F_n = \frac1n \cdot 1_{[0,n]}(x)$
we know that $\int F_n \, d\mu = 1 \text{ and not $\int0 \, d\mu$}$