What's the relationship between continuity property of Lebesgue measure and continuity on a metric space? This is a topic from Lebesgue measure in $\textit {Carothers' Real Analysis}$:


I know how to prove Theorem 16.23. However, I can not figure out why he names this property as continuity? Besides what's the relationship between continuity here and continuity on a metric space?(I mean it is a little bit weird to say continuous Lebesgue measure)
Some definitions on the book:


*

*Continuity properties on a metric space:


*Lebesgue outer measure:


*Capital $M$ is introduced here:


*Lebesgue measure:


Not clearly. Continuity property of Lebesgue measure is just a nature term defined that measure of a limit is equal to limit of the measure. And with this, we can take limit in parentheses out (like continuity in a metric space that $f$ is continuous <=> $limit_{n->+∞} f(x_n) = f(limit_{n->+∞} x_n$) whenever $limit_{n->+∞} x_n = x$).
