A general solution of the functional equation \(\int^{\infty}_{0}f(x+y)d\mu (y)=f(x)\) where f is a nonnegative function and \(\mu\) is a \(\sigma\)-finite positive Borel measure on [0,\(\infty)\) is shown to be \(f(x)=p(x)\exp (\lambda x)\) where p is a periodic function with every \(y\in {\underline \mu}\), the support of \(\mu\), as a period.
The solution is applied in characterizing Pareto, exponential and geometric distributions by properties of integrated lack of memory, record values, order statistics and conditional expectation.