Let $(X,\Sigma,\mu)$ be a measure space and consider a family of $\mu$-measurable functions $f_i:X \to \mathbb{R}$ for $i$ lying in some index set $I$. Define $$f(x) = \inf_{i \in I} f_i(x)$$

I think every good analysis book mentions or proves that if $I$ is countable, then $f$ is also $\mu$-measurable. What is not so clear is:

If $I$ has cardinality of the continuum, is $f$ still $\mu$-measurable?

Since I strongly suspect that the answer is "no" although a counterexample is not immediately coming to mind, here is the real question:

What are minimal conditions on $f_i$ that will make $f$ $\mu$-measurable even when $I$ is uncountably large?

By minimal conditions I am hoping for some weak properties, such as those that hold $\mu$-almost everywhere rather than those that require something strong from the global structure of $I$, such as a (partial or total) ordering.

If you know of a book or paper that deals with this, please let me know. I understand there is a good chance this type of thing is not considered research material by folks here; in this case I will delete the question.