Good lambda inequalities for non-doubling measures in $\mathbb{R}^n$

Abstract: We establish a good lambda inequality relating to the distribution function of Riesz potential and fractional maximal function on $\left(\mathbb{R}^n, d\mu\right)$ where $\mu$ is a positive Radon measure which doesn't necessarily satisfy a doubling condition. This is extended to weights $w$ in $A_{\infty}(\mu)$ associated to the measure $\mu$. We also derive potential inequalities as an application.