= Solution
{c}
Use the characterization of <measurability> by <preimages> of open half-lines, allowing intermediate extended-real-valued functions. Put $u_m=\inf_{n\geq m}f_n$. For each real $a$,
$$
\{u_m<a\}=\bigcup_{n\geq m}\{f_n<a\}\in\Sigma.
$$
Thus every $u_m$ is measurable. Countable suprema are measurable too, since $\{\sup_m u_m>a\}=\bigcup_m\{u_m>a\}$ and the upper half-lines also generate the <Borel sigma-algebra>. Therefore
$$
\boxed{\liminf_n f_n=\sup_m\inf_{n\geq m}f_n}
$$
is measurable. Its values are real by the stated hypothesis, so it is the required ordinary real-valued <measurable function>, rather than only an extended-real-valued one.
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