This Corollary follows from proposition 2.7:
If $\{f_j\}$ is a sequence of $\overline{\mathbb{R}}$-valued measurable functions on $(X,M)$, then the functions $$\begin{aligned} g_1(x) = \sup_{j}f_j(x), \ \ \ \ g_3(x) = \lim_{j\rightarrow \infty}\sup f_j(x) \end{aligned}$$ $$\begin{aligned} g_2(x) = \inf_{j}f_j(x), \ \ \ \ g_4(x) = \lim_{j\rightarrow \infty}\inf f_j(x) \end{aligned}$$ are all measurable functions. If $f(x) = \lim_{j\rightarrow \infty}f(x)$ exists for every $x\in X$, then $f$ is measurable.
If you want a proof for this let me know I will post it.
Corollary 2.8: If $f,g:X\rightarrow \overline{\mathbb{R}}$ are measurable, the so are $\max(f,g)$ and $\min(f,g)$
Proof (suggested from CopperHat user): Suppose we have $f,g:X\rightarrow \overline{\mathbb{R}}$ are measurable, define $f_k = f$ for odd $k$ and $f_k = g$ for even $k$. So, if $f < g \Rightarrow \sup_k f_k(x) = f$ and if $f > g \Rightarrow \sup_k f_k(x) = g$. So then, $\max(f,g) = g_1$. The same argument can be applied for $\min(f,g)$, hence $\max(f,g)$ and $\min(f,g)$ are measurable.