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In this two-paper series, we prove the invariance of the Gibbs measure for athree-dimensional wave equation with a Hartree nonlinearity. The main noveltyis the singularity of the Gibbs measure with respect to the Gaussian freefield. The singularity has several consequences in both measure-theoretic anddynamical aspects of our argument.  In this paper, we construct and study the Gibbs measure. Our approach isbased on earlier work of Barashkov and Gubinelli for the $\Phi^4_3$-model. Mostimportantly, our truncated Gibbs measures are tailored towards the dynamicalaspects in the second part of the series. In addition, we develop new toolsdealing with the non-locality of the Hartree interaction. We also determine theexact threshold between singularity and absolute continuity of the Gibbsmeasure depending on the regularity of the interaction potential.

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures