In the context of continuity of probabilities we define $\{A_n\}\nearrow A$ to mean that $A_1 \subseteq A_2 \subseteq A_3 \subseteq ...$ and $\cup_n A_n=A$, where $A, A_1, A_2, A_3, ... \in \mathcal{F}$ for some probability triplet $(\Omega, \mathcal{F}, P)$. I have no problems with this notation.
But how is $\{f_n(x)\}\nearrow x$ interpreted, where $f_n:[0,\infty) \rightarrow [0,\infty)$ and $x$ is a random variable? I don't see how $f_1(x) \subseteq f_2(x) \subseteq f_3(x) \subseteq ...$ considering $f_n(x)$ is not a set but a real value. A possible interpretation might be that it's range of $f_n$ that is a subset of the range of $f_{n+1}$; but then it's unclear how $\cup_n f_n(x)=x$, i.e. how the union of ranges would correspond to a real value.
(Apparently $\{f_n(x)\}\nearrow x$ is useful when deriving the expectation of general non-negative random variables.)
It means that $f_n(x)\le x$ for all $n,$ and that $\lim_{n\to\infty}f(x)=x.$ Verbally, we may say that $f_n(x)$ converges to $x$ from below. Often, it also indicates that $f_n(x)$ is monotonically increasing, but this varies from source to source.