Almost Sure Modifications of Gaussian Random Vectors are Gaussian

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $d$ be a natural number, let $(Y_1,\dots,Y_d)$ be a Gaussian random vector on $(\Omega,\mathcal{F},P)$, and let $X_1,\dots,X_d$ be random variables on $(\Omega,\mathcal{F},P)$ with

here each $\{X_i=Y_i\}$ is an event, since differences of random variables are random variables (as noted in Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process) and $\{0\}$ is a Borel set; thus each $X_i$ equals $Y_i$ almost surely.

Then $(X_1,\dots,X_d)$ is a Gaussian random vector; every Gaussian representation of $(Y_1,\dots,Y_d)$ is a Gaussian representation of $(X_1,\dots,X_d)$; and the mean vectors and covariance matrices of the two tuples coincide: