The Mollified Density Cost of a Probability Measure with Finite Second Moment

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d$ be a natural number with $1\le d$, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and let $\mathcal{P}_{2}(\mathbb{R}^{d})$ be the set of probability measures with finite second moment. Let $\eta$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$ and let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$; for $\mu\in\mathcal{P}(\mathbb{R}^{d})$, $\eta_{\varepsilon}*\mu:\mathbb{R}^{d}\to\mathbb{R}$ is the mollified density of $\mu$, formed with the rescaled kernel $\eta_{\varepsilon}$ as in Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost. Let $L\in\mathbb{R}$ be nonnegative and let $\Phi$ be a convex Lipschitz integrand with constant $L$.

(Mollified density cost)¶ For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, the mollified density cost of $\mu$ with integrand $\Phi$, kernel $\eta$ and scale $\varepsilon$ is the real number

the integral of the function $\Phi\circ(\eta_{\varepsilon}*\mu)$, which is Borel and integrable with respect to $\lambda_{d}$ by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §cost. The kernel $\eta$ is suppressed from the notation.