In the earlier post , 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.
- 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 .
- 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.
- If is measurable and is a countable set, then is measurable.
This claim follows from
i.e. Borel -algebra is closed under countable union.