Reverse Hardy-type inequalities for supremal operators with measures

In this paper we characterize the validity of the inequalities $\|g\|_{p,(a,b),\lambda} \le c \|u(x) \|g\|_{\infty,(x,b),\mu}\|_{q,(a,b),\nu}$ and $\label{eq.0.1.2} \|g\|_{p,(a,b),\lambda} \le c \|u(x) \|g\|_{\infty,(a,x),\mu}\|_{q,(a,b),\nu}$ for all non-negative Borel measurable functions $g$ on the interval $(a,b)$, where $0 < p \le +\infty$, $0 < q \le +\infty$, $\lambda$, $\mu$ and $\nu$ are non-negative Borel measures on $(a,b)$, and $u$ is a weight function on $(a,b)$.