Let $\{h_n\}$ be a sequence of extended real valued functions. In Lemma 3.5 of Probabilities with Martingales Williams uses the notations $\inf h_n$ and $\liminf h_n$ in such a way as to imply these two mean different things. My first thought when reading these expressions (knowing they refer to functions) is that $$\inf h_n:x\mapsto \inf\{h_n(x) : n\in\mathbb{N}\} \ \ \ \ \text{ and } \ \ \ \ \liminf h_n : x\mapsto \liminf_{n\to\infty} h_n(x),$$ but then -if I'm not mistaken- both functions would be equal.
In the lemma, Williams shows that if the $\{h_n\}$ are measurable, then so are $\inf h_n$ and $\liminf h_n$. His proof that $\lim h_n$ is measurable makes sense if I am to interpret the expression as I have explained above, but that leaves me unsure as to what $\liminf h_n$ means, even after reading (although not understanding) the proof that such function is also measuable.
What is the meaning of $\ \liminf h_n$?
The definition seems quite clear to me: $$\begin{align} \inf h_n\colon X &\to \overline{\mathbb{R}}\\ x &\mapsto \inf\{h_n(x)\mid n\in \mathbb{N}\}\end{align}$$ and $$\begin{align}\liminf h_n\colon X&\to \overline{\mathbb{R}}\\ x&\mapsto \sup\big\{\inf\{h_r(x)\mid r\geq n\}\big|n\in\mathbb{N}\big\}\end{align}$$
Let $h_n(x)=1$ for all $n\neq 0$ and let $h_0(x)=0$. Then $\forall x\in X,\; \inf h_n(x)=0$ and $\liminf h_n(x)=1$ and thus $\liminf\neq \inf$ (in fact, $\inf h_n\leq \liminf h_n$ is always true).
Maybe a more useful way of defining $\liminf$ (which is equivalent) is by $$\liminf_n h_n=\lim_n\inf\{h_r\mid r\geq n\}.$$