Given a measure space (X, F, mu), $f_n$'s are given to be non negative measurable. $f_n$ decreases pointwise to $f$ and $f_1$ is given to be integrable. Then $\int f=\int f_n$
Proof: Let $g_n:=f_1-f_n$. Then $g_n$'s are non negative and increasing pointwise to $f_1-f$. So by monotone convergence theorem, $\int f_1-f_n\uparrow \int f_1-f$