Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content

Let $E\subseteq\mathbb R^{p+q}$ be a bounded Jordan set whose sections $E_x$ are Jordan measurable outside a content-zero set of parameters. Then the completed sectional-content function $x\mapsto\operatorname{cont}(E_x)$ is integrable and $\operatorname{cont}(E)=\int\operatorname{cont}(E_x)\,dx,$ with empty sections assigned content $0$.

Consequently, if bounded Jordan sets $E,F\subseteq\mathbb R^{p+q}$ have Jordan sections outside content-zero exceptional parameter sets and $\operatorname{cont}(E_x)=\operatorname{cont}(F_x)$ wherever both are ordinary Jordan sections outside those sets, then $\operatorname{cont}(E)=\operatorname{cont}(F)$.