  In this two-paper series, we prove the invariance of the Gibbs measure for a
three-dimensional wave equation with a Hartree nonlinearity. The main novelty
is the singularity of the Gibbs measure with respect to the Gaussian free
field. The singularity has several consequences in both measure-theoretic and
dynamical aspects of our argument.
  In this paper, we construct and study the Gibbs measure. Our approach is
based on earlier work of Barashkov and Gubinelli for the $\Phi^4_3$-model. Most
importantly, our truncated Gibbs measures are tailored towards the dynamical
aspects in the second part of the series. In addition, we develop new tools
dealing with the non-locality of the Hartree interaction. We also determine the
exact threshold between singularity and absolute continuity of the Gibbs
measure depending on the regularity of the interaction potential.
