Diagonal Gaussian Measures on Euclidean Space

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$. Let $c$ be a variance vector and $\rho_{c}$ the diagonal Gaussian density with variances $c$, which is positive and Borel by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity.

(Diagonal Gaussian measure)¶ The diagonal Gaussian measure with variances $c$ is the measure with density $\rho_{c}$ with respect to $\lambda_{d}$, written $\gamma_{c}$; that is, $\gamma_{c}(B)=\int_{\mathbb{R}^{d}}\mathbf{1}_{B}\,\rho_{c}\,d\lambda_{d}$ for $B\in\mathcal{B}(\mathbb{R}^{d})$. Since $\mathbf{1}_{\mathbb{R}^{d}}\rho_{c}=\rho_{c}$, this formula with $B=\mathbb{R}^{d}$ and The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §normalization give

$$\gamma_{c}(\mathbb{R}^{d})=\int_{\mathbb{R}^{d}}\mathbf{1}_{\mathbb{R}^{d}}\,\rho_{c}\,d\lambda_{d}=\int_{\mathbb{R}^{d}}\rho_{c}\,d\lambda_{d}=1,$$

so $\gamma_{c}$ is a probability measure on $\mathbb{R}^{d}$.