Monotone convergence when $f_n$ is decreasing I know that if $f_n\geq 0$, that $(f_n)$ is increasing and that if $\lim_{n\to \infty }f_n=f\in L^1$, then,
$$\lim_{n\to \infty }\int f_n=\int f.$$
Does this result also work when $(f_n)$ is decreasing ? 

Let $f_{n}=\frac{1}{n}χ[0,n]$. 
For every $\epsilon > 0$ and $x \in \mathbb{R}$ there exists $N > 1$
 such that
$|f_n(x)| < \epsilon $ for every $n>N$. Hence, $f_n$ converges to $f = 0$ ,uniformly. However, 
$$0 = \int
f d\lambda \not= \lim \int
f_n d\lambda = 1$$
The MCT does not apply because the sequence is not monotone increasing. Fatou’s
lemma obviously applies.
