In this paper, we essentially characterize the real power of the convolution set of X + Y, where X and are two independent random variables which have respectively a Bernoulli distribution, with parameter and an infinitely divisible probability distribution ν. The above problem is equivalent to finding the set of such that the mapping is a Laplace transform of some probability distribution. This class of real power of convolution x is provided and described. The obtained results generalize the case where Y is Gamma or Negative Binomial distributed.