limit of integrals of increasing functions Let $\{f_n\}$ be a non-decreasing sequence of integrable functions such that $\lim_{n \to \infty} f_n \geq 0$. If $S = \{x|\lim_{n \to \infty}f_n > 0\}$ is not a null set, show that $\lim_{n \to \infty} \int f_n > 0$.
I started by picking a large $N$ where $\{x | f_n(x) \geq 1/N\}$ is not null, and then, using Fatou's lemma, on this set only, I can show the integral is positive.
How do I extend to the full integral?
EDIT: The problem as it was originally stated is incorrect. Corrected.
 A: First, note that since the sequence of measurable functions $(f_n)$ is non-decreasing, it follows that $\lim_{n\rightarrow+\infty}f_n$ is a well-defined measurable non-negative function. Its integral is thus well-defined.
The sequence $(f_n-f_1)_{n\in\mathbb N}$ is nonnegative, so Fatou's lemma applies:
$$
\liminf_{n\rightarrow+\infty}\int \left(f_n-f_1\right)\ge\int\lim_{n\rightarrow+\infty}\left(f_n-f_1\right).
$$
Additionally, $\liminf_{n\rightarrow+\infty}\int \left(f_n-f_1\right)=\left(\liminf_{n\rightarrow+\infty}\int f_n\right)-\int f_1$ (because both $f_n$ and $f_1$ are integrable). Since $\int|f_1|<+\infty$, it thus holds that
$$
\liminf_{n\rightarrow+\infty}\int f_n\ge\int\lim_{n\rightarrow+\infty}f_n.
$$
As mentioned in OP, take $N\in\mathbb N$ sufficiently large, so that $S_N=\{x\ :\ \lim_{n\rightarrow+\infty}f_n(x)\ge1/N\}$ is such that $=\mu(S_N)>0$, and note that since $\lim_{n\rightarrow+\infty}f_n$ is nonnegative,
$$
\int\lim_{n\rightarrow+\infty}f_n\ge\int\lim_{n\rightarrow+\infty}f_n1_{S_N}\ge\frac1N\mu(S_N)>0.
$$
Hence,
$$
\liminf_{n\rightarrow+\infty}\int f_n>0.
$$
Lastly, since the sequence of integrals $(\int f_n)_{n\in\mathbb N}$ is non-decreasing (since $(f_n)$ is) then the sequence of integrals converges (or tends to $+\infty$), and 
$$
\lim_{n\rightarrow+\infty}\int f_n>0.
$$
A: There might be something missing with the statement, take $f_n: [0,1]\rightarrow \mathbb{R}$ with $n\geq 2$
$$
   f_n(x) = \left\{
     \begin{array}{lr}
       -n & : x \in (0,\frac{1}{n}]\\
       2 & : x \in (\frac{1}{2}, 1]\\
       0 &  \text{otherwise}
     \end{array}
   \right.
$$
each $f_n$ is non-decreasing with non-negative pointwise limit, $\mu(S) = \frac{1}{2}$, but $\int f_n = 0$ for all $n$.
Edit: If $\{f_n\}$ is a non-decreasing sequence of integrable functions, we have
$$\lim_n \int f_n = \int \lim_n f_n \geq \epsilon \mu(\{x : f(x) \geq \epsilon\})> 0$$
for some $\epsilon > 0$. The equality is due to Lebesgue dominated convergence theorem with the bound $|f_1| + |\lim_n f_n|$, the inequality is due to Markov inequality for non-negative functions.
