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:



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> If $I$ has cardinality of the continuum, is $f$ still $\mu$-measurable?
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Since I strongly suspect that the answer is "no" although a counterexample is not immediately coming to mind, here is the real question:



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



When is the infimum of an arbitrary family of measurable functions also measurable?
As you expected, the infimum of continuum many measurable functions need not be measurable, even in the case where $X$ is the real line with Lebesgue measure. In fact, if $A$ is any subset of $\mathbb R$ (in particular not necessarily measurable), its characteristic function is the infimum of at most continuum many measurable functions, namely the characteristic functions of the sets $\mathbb R-\{x\}$ for all $x\notin A$. 


A similar argument shows that even the infimum of $\aleph\_1$ measurable functions won't be measurable if, in your measure space, the intersection of some $\aleph\_1$ measurable sets is not measurable. And there are such measure spaces. For example, take $X$ to be a set of cardinality $\aleph\_1$, take $\Sigma$ to be the $\sigma$-field of countable and co-countable sets, and take $\mu$ to give the countable sets measure 0 and the co-countable sets measure 1.


Whether the same thing happens for Lebesgue measure on $\mathbb R$ is independent of the usual ZFC axioms of set theory. It is consistent with ZFC that there is a non-Lebesgue-measurable set of cardinality $\aleph\_1$, and it is also consistent that every union of $\aleph\_1$ sets of Lebesgue measure 0 has measure 0, which implies that every union or intersection of $\aleph\_1$ measurable sets is measurable. 


The smallest cardinal $\kappa$ such that the union of some $\kappa$ sets of Lebesgue measure 0 does not have measure 0 is called the *additivity* of Lebesgue measure; it is one of the many well-studied cardinal characteristics of the continuum. It is also the smallest $\kappa$ such that the infimum of some $\kappa$ Lebesgue measurable functions $\mathbb R\to\mathbb R$ fails to be measurable.


