# fa.functional analysis – "almost" absolute continuity of the derivation of the BV function if \$ { rm Tr} , D_Sf = 0 \$

To let $$f: mathbb R ^ N to mathbb R ^ N$$ be a $$BV$$ Function.
Suppose that $$mathrm {div} f$$ is absolutely continuous with respect to the Lebesgue measure: $$operatorname {div} f ll mathcal L ^ N$$, This implies, as shown in a related question,
The $${ rm Tr} , D_Sf = 0.$$

Does that mean that? $$D_S f$$ is nearly absolutely uninterrupted in a sense? How can one properly formulate this notion of "almost" absolute continuity?

Here is a more specific question:

• As mentioned in the question Lusin Lipschitz approximation in BV and Sobolev room, $$f$$ Lipschitz is outside a small amount (small in relation to the Lebesgue measure). does $${ rm Tr} D_S f = 0$$ imply that this quantity is also "small" in relation to the singular measure $$D_S f$$?

Related questions are placed in the BV function with absolutely continuous divergence and role of the absolute continuity of the divergence of the BV function to demonstrate renormalization properties