  We introduce a notion of weak solution of the master equation without
idiosyncratic noise in Mean Field Game theory and establish its existence,
uniqueness up to a constant and consistency with classical solutions when it is
smooth. We work in a monotone setting and rely on Lions' Hilbert space
approach. For the first-order master equation without idiosyncratic noise, we
also give an equivalent definition in the space of measures and establish the
well-posedness.
