2
$\begingroup$

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 $\int0 \, d\mu$}$

  • 1
    The fact that $(f_n)$ is non-increasing is not a big problem in itself since you could consider $-f_n$ and $-f$. The reason why the monotone convergence theorem doesn't apply here is that it needs $-f_n$ to be **non-negative**, i.e. $f_n$ non-positive.2012-12-06

1 Answers 1

6

Why not $f_n = 1_{[n,\infty)}$ and $f = 0$ ?

  • 0
    Yes, for example.2012-12-06