I am interested in understanding the set $$ A := \lbrace x \in \mathbb{R}: f(x) \leq 0\rbrace $$ where $f \in BV(\mathbb{R})$. I know that there is a decomposition of the variational measure in jump part, Cantor part and continuous part. Is there a similar characterization for the set $A$? Can sublevel sets of $BV$ functions be "arbitrarily ugly" or can they only be, say a union of intervals and cantor sets? Also (this might be obvious), with $BV$ functions I mean $L^1$-functions and not pointwise $BV$, so $\chi_{\mathbb{Q}} = 0$ for example.
Is there any literature on this or could you maybe provide a reference? Thank you in advance. Also I don't mind receiving some references on the decomposition result for the measure.
I also know Leoni's and Evans'/Gariepy's books on this, which did not really help me (maybe I missed something). Unfortunately, I don't have access to Ambrosio's book.