This result comes after establishing the countable additivity of measurable sets in Stein and Shakarchi (2009).
Suppose $E1,E2,...$ are Lebesgue-measurable subsets of $\mathbb{R}^d$. If $(E_k)\uparrow E$, then we have $m(E)=\lim_{N\rightarrow\infty}m(E_N)$.
Why is this result useful or perhaps significant? Doe this result introduces the notion of continuity in the world of Lebesgue measurable sets?
Reference: $\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.