The following equation \(f(x)=\int^{\infty}_{0}f(x+y)d\mu (y),\quad a.e.\quad x\geq 0,\) where \(\mu\) is a positive regular Borel measure defined in (0,\(\infty)\), called an integrated Cauchy functional equation, is discussed in order to establish its relevant nonnegative locally integrable solutions in (0,\(\infty)\).