 BV slicing theorem

For a BV space function $a$, almost every restriction $a_y$ to a line parallel to $e_j$ has bounded variation on the corresponding domain slice. Directional derivative variation disintegrates as $|D_{j}a|={\mathcal{L}}^{n-1}\otimes |D{a}_y|$. Slice jump values agree with the appropriate oriented BV traces on a hypersurface for almost every crossing of a rectifiable jump surface. Integrating slice jump sums produces a surface integral with projection factor $|\nu \cdot e_j|$. Uniform essential bounds on the slices are not asserted for an unbounded function.