Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$ I need your help to understand and analyse the following problem:
Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu =  \int_X  \lim_k f_k d\mu$.
Is $f^-$ here the second part of $f_k = f_k^+ -f_k^-$? And how we can think about this problem? How we can use this fact? Do we have to use Fatou's Lemma or Monotone Convergence Theorem? But the sequence may or may not have a limit (nothing mentioned about it's limit), so probably we may use Fatou's, since it does not require limit to exist.
Or do you think $-$ on $f_1$ is a typo? It might be $f_1'$ or $f_1$. If so, then why?
Any hints or possible solutions?

The statement is a generalization of Monotone Convergence Theorem : 
Consider sequence of functions $g_n(x) = f_n(x)+f_1^-(x)$ (Here $f_1^-$ is understood in the sense $f_1 = f_1^+ - f_1^-$)
Now, observe that sequence $g_n$ is nondecreasing and $g_n \geq 0$ for every $n$. So, from MCT
$$\lim_n\int g_n d\mu = \int \lim_n g_n d\mu $$ which means
$$\lim_n \int f_n d\mu + \int f_1^- d\mu = \int \lim_n f_n d\mu + \int f_1^- d\mu$$
and as $\int f_1^- d\mu < \infty$ we can cancel $\int f_1^- d\mu$ on both sides of the above equality and hence we have the result!
