If $\{f_n\}\subset L^+, f_n$ decreases pointwise to $f,$ and $\int f_1<\infty,$ then $\int f=\lim\int f_n$

Question

If $\{f_n\}\subset L^+, f_n$ decreases pointwise to $f,$ and $\int f_1<\infty,$ then $\int f=\lim\int f_n.$

I am not really sure why we need the integration of f1 finite. I am trying to use Monotone Convergent Theorem, but in this one, fn is decreasing though.

Answer 1

Each $f_{n}$ is dominated by $f_{1}\in L^{1}$, so apply the dominated convergence theorem