Lebesgue outer measure인 경우에는 continuity from below가 measurability에 상관없이 항상 성립한다.

Problem.If , then

Proof. If , then . So we may assume that . Let be a measurable hull of and define .

Then note that is also a measurable hull of since for any . So .

Also,

Also, …

So by continuity of measure, we have

Then by fact, we have

since is a measurable hull of and is a measurable hull of