Equivalence of some subcritical properties in continuum percolation

We consider the Boolean model on Rᵈ. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let C denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that E(|C|) is finite if and only if there exists $A,B>0$ such that P(≥ n)≤ Ae-Bn for all n≥1.