A coordinate shear preserves Jordan content by translating every section

The shear matrix is the identity with one off-diagonal entry $c$, so its determinant is $1$. By [L2] the linear image $S(E)$ is Jordan measurable and $\operatorname{cont}(S(E))=|1|\operatorname{cont}(E)=\operatorname{cont}(E)$.

Sections show the same thing wherever [L1] applies. Hold all coordinates except $x_j$ fixed: the corresponding section of $S(E)$ is the section of $E$ translated by $cx_i$, so its one-dimensional content is unchanged, and for a set whose sections are Jordan outside a content-zero parameter set [L1] integrates these equal values to the same total. This is a second reading of the result and not a second proof of it: a bounded Jordan set need not have Jordan sections outside a content-zero parameter set, so [L1] is not available for an arbitrary $E$ and step 1.1 carries the statement.