A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure.

A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesgue integration theory, we rarely use upper Lebesgue integral.

Why are outer measure and lower integral more important than inner measure and upper integral?

whythe Caratheodory definition seems to be presented more frequently. $\endgroup$ – Noah Schweber Aug 22 '18 at 2:41