Monotone convergence theorem without monotonicity

Question

I have a question related to the monotone convergence theorem. Consider a situation in which all assumptions of the monotone convergence theorem are satisfied except the monotonicity, i.e. we have a sequence of measurable positive functions $f_k:\mathcal{X} \in \mathbb{R}\rightarrow [0,\infty)$ such that $\lim_{k \rightarrow \infty}f_k=f$ $\forall x$, but which is not necessarily monotone increasing. Is there a way to apply the monotone convergence theorem without assuming $f_k\leq f$ $\forall k$?

Answer 1

No. For a counterexample (to equality for the sequence of integrals) and the proper tool to use in this case look at Fatou's lemma

Answer 2

No.

Consider this example. Let $X=(0,1)$ and $$f_k(x)=\left\{\begin{array}{ll}k^2&\mathrm{if}\,\,x\in(0,\frac{1}{k})\\0&\mathrm{otherwise}\end{array}\right.$$ Then $f_k\rightarrow 0$ a.s., $\lim_k\int f_k\mathrm{d}m=\infty\neq 0=\int f\mathrm{d}m$.