The ${\mathrm{\Phi }}_3^4$ measure via Girsanov's theorem

We construct the ${\mathrm{\Phi }}_3^4$ measure on a periodic three dimensional box as an absolutely continuous perturbation of a random shift of the Gaussian free field. The shifted measure is constructed via Girsanov’s theorem and the relevant filtration is the one generated by a scale parameter. As a byproduct we give a self-contained proof that the ${\mathrm{\Phi }}_3^4$ measure is singular wrt. the Gaussian free field.