TITLE: Scheffe’s Theorem
QUESTION [7 upvotes]: I saw a statement of Scheffe’s Theorem as follows ([1, p84]):

... we need a simple result called Scheffe’s Theorem. Suppose we have probability densities $f_n$ , $1 \le n \le \infty$, and $f_n \to f_\infty$ pointwise as $n \to \infty$. Then for all Borel sets $B$
  $$
\left| \int_B f_n (x)dx - \int_B f_\infty(x) \, \mathrm dx~\right| \le
\int_B |f_n(x) − f_\infty(x)| \, \mathrm dx
=\int 2 (f_\infty(x) − f_n(x))^+ \, \mathrm dx \to 0
$$
  by the dominated convergence theorem, the equality following from the fact that the
  $f_n ≥ 0$ and have integral $= 1$.

I feel that to use dominated convergence theorem, and to make the last equality work, we need one more condition that $\int f_\infty(x) \, \mathrm dx < \infty$. Or in this case actually this condition is not necessary?

REPLY [11 votes]: The condition that $\int f_\infty < \infty$ can be deduced from the other assumption. By assumption, we have $f_n \to f_\infty$ pointwise and each $f_n$ is a density. Hence $\int f_n = 1$ for all $n$. From Fatou's lemma it follows that
$$\int f_\infty = \int \liminf f_n \le \liminf \int f_n = 1.$$