Are non-increasing absolutely continuous functions dense in non-increasing, $L^1([a,b])$ functions?

Question

I'd like to know whether the set of non-increasing absolutely continuous function is dense in the space of non-increasing functions in $L^1([a,b])$, equipped with the metric $d(f,g)=\int_{[a,b]}\vert f(x)-g(x)\vert dx$.

Most proofs I have seen about density in $L^1$ involves using polynomials but they are not necessarily non-increasing, and it is unclear to me how to modify them to preserve density.

Any hint or reference is welcomed.