Renewal measure ($U={\sum }_{j\ge 0}{K}^{\ast j}$)

For a positive interarrival law $K$, the renewal measure $U={\sum }_{j\ge 0}{K}^{\ast j}$ counts the expected number of renewal epochs in a set, including the epoch at zero through $K^{\ast 0}={\delta }_0$. Its cumulative function is one plus the usual renewal function. Convolution against $U$ solves a renewal equation.