Consider a non-negative Lebesgue-integrable function $f : X \rightarrow \mathbb{R}$, where $X$ is a measure space, and let $F = \{(x,y)|x \in X, y \in [0,f(x)]\}$. Can the Lebesgue integral of $f$ be interpreted as $\eta(F)$, where $\eta$ is some sort of inner measure induced on the set $X \times \mathbb{R}$?
Rationale: When defining the Lebesgue-integral, one speaks of "horizontal strips" that are subsets of $F$. Thus, it looks like an inner measure.