\begin{definition}\label{phi-min-def}\lean{phiMinimizes}\leanok  Given $G$-valued random variables $X,Y$, define
$$ \phi[X;Y] := d[X;Y] + \eta(\rho(X) + \rho(Y))$$
and define a \emph{$\phi$-minimizer} to be a pair of random variables $X,Y$ which minimizes $\phi[X;Y]$.
\end{definition}
def phiMinimizes {Ω : Type*} [MeasurableSpace Ω] (X Y : Ω → G) (η : ℝ) (A : Finset G)
    (μ : Measure Ω) : Prop :=
  ∀ (Ω' : Type uG) (_ : MeasureSpace Ω') (X' Y' : Ω' → G),
    IsProbabilityMeasure (ℙ : Measure Ω') → Measurable X' → Measurable Y' →
    phi X Y η A μ ≤ phi X' Y' η A ℙ