Lower semicontinuity for an integral functional in BV

We prove a lower semicontinuity result for a functional of linear growth initially defined by $${\int }_{\mathrm{\Omega }}F\left(\frac{dDu}{d\mu }\right)d\mu$$ for $u\in BV(\mathrm{\Omega };{\mathbb{R}}^N)$ with $Du\ll \mu$. The positive Radon measure $\mu$ is only assumed to satisfy ${\mathcal{L}}^n\ll \mu$.