The paper introduces a new type of grand Lebesgue space, for which special cases are weighted grand Lebesgue spaces introduced by Fiorenza, Gupta and Jain (2008) and a generalization of grand Lebesgue spaces introduced by Greco, Iwaniec and Sbordone (1997). The main result is a mean ergodic theorem, in the Von Neumann sense, for some operator acting on the closure of the set of compactly supported in the newly introduced grand Lebesgue space.