Suppose $X$ is a multivariate normal distribution in $\mathbb{R}^3$ with a covariance matrix whose rank is 2. Therefore, it is a singular distribution. Is it possible to represent it as a push forward of some non-singular multivariate normal distribution in $\mathbb{R}^2$? If the answer is yes, how to define the measurable map $f:\mathbb{R}^2\rightarrow \mathbb{R}^3$ that defines the push forward, that is, $\forall B \subset \mathbb{R}^3, \mu_3(B)=\mu_2(f^{-1}(B))$, where $\mu_2$ and $\mu_3$ are respective Gaussian measures in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively?
$\textbf{EDIT}:$ I am editing the question to give an example. As per the answer of @sean507, yes I believe PCA or singular value decomposition (SVD) is the algebraic answer. But I am trying to understand how to write the expectation in a technically correct way in terms of measure theory. Let me present a simple example. Suppose the multivariate normal distribution has mean $\mu=(\mu_1,\mu_2,\mu_3)\in \mathbb{R}^3$ and the covariance matrix $\Sigma\in\mathbb{R}^{3\times 3}$ is diagonal with the diagonal entries: $\sigma_1>\sigma_2>\sigma_3=0$. Now assume $a=(a_1,a_2,a_3)\in\mathbb{R}^3$ to be a fixed vector. I want to find the expected value of $a^{\top}X$ where $X\sim N(\mu,\Sigma)$. If I can sample $x_1,x_2,\ldots,x_n\in\mathbb{R}^3$ i.i.d. from $N(\mu,\Sigma)$ then for large enough $n$, the expected value can be estimated reasonably well as, $\frac{1}{n}\sum_{i=1}^na^{\top}x_i$. Any such $x_i$ can be sampled as follows: sample its first coordinate from $N(\mu_1,\sigma_1)$, sample its second coordinate from $N(\mu_2,\sigma_2)$ and set the third coordinate to zero. But how do we write this expectation, whose numerical value should be $a_1\mu_1+a_2\mu_2$, in terms of integral in a technically correct measure theoretic way since the density of $X$ is not defined as it is a singular distribution? Here both $a$ and $X$ are in $\mathbb{R}^3$, even though $X$ is embedded (and supported) on a coordinate plane in $\mathbb{R}^2$ which has a Lebesgue measure zero in $\mathbb{R}^3$.
Further, let $\pi_2$ be the coordinate projection onto the first two coordinates of any vector in $\mathbb{R}^3$. Assume $a'=\pi_2(a)$ and $X'$ to be a multivariate normal distribution with mean $\mu'=\pi_2(\mu)$ and covariance matrix $\Sigma'\in\mathbb{R}^{2\times 2}$ which is a diagonal matrix with diagonal entries $\sigma_1>\sigma_2$. $X'$ is a non-singular distribution and the expected value $\mathbb{E}(a'^{\top}X')$ is well defined (in a measure theoretic way) which is same as the expected value of $a^{\top}X$. I am just trying to understand how to write the expected value of $a^{\top}X$ in a technically correct measure theoretic way.
