Ref:

[1] Supremum of a class of functions

In the earlier post [1], we discussed measurability of the infimum of a class of measurable functions. In particular, for the infimum of a class of measurable functions as a function, we can show that it may not be measurable. Therefore, we shall need additional conditions to have the infimum function measurable. In this post, we show that the infimum function is

- measurable if the class size is countable;
- lower semicontinuous (thus measurable) if each function in the class is lower semicontinuous.

For simplicity, we consider the real number space , but all discussions below extend to a complete metric space. One can use all open balls to generate the smallest -algebra, which is referred as Borel -algebra, denoted by . A set in is called a Borel set.

Given , the infimum function is defined by, for some set

Our question is when is (Borel) measurable? Recall is measurable, if , i.e.

In general,

- even if is jointly measurable, i.e. , the infimum may not be measurable.

Indeed, we can consider the following example: Let and . Then is measurable, since . But is not measurable, since .

However,

- If is continuous for each , then is upper semicontinuous, hence is measurable.

To see that, we write for any

Union of possibly uncountably many open sets is open, and is open. So is USC.

Alternatively,

- If is measurable and is a countable set, then is measurable.

This claim follows from

i.e. Borel -algebra is closed under countable union.

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