I would like to check if I am correct about my expression.
Let $d\mu$ be a Gaussian measure on $\mathbb{R}^n$ with mean $\beta \in \mathbb{R}^n$ and covariance matrix $B$ given by
\begin{equation} B=\alpha \alpha^T \end{equation} for some fixed nonzero column vector $\alpha \in \mathbb{R}^n$.
Then, it is well-known that $d\mu$ is degenerate for $n>1$ and its support is given by the affine line \begin{equation} \bigl[ \beta + c \alpha \mid c \in \mathbb{R} \bigr] \subset \mathbb{R}^n \end{equation}
Now, I would like to use the parameter $c$ to express the density function of this Gaussian measure $d\mu$. Then is it correct that \begin{equation} d\mu\Bigl(\beta + c \alpha \Bigr)= \frac{1}{\sqrt{2\pi}}e^{-\frac{c^2}{2}}dc \end{equation} where $dc$ is just the Lebesgue measure on $\mathbb{R}$.
I am aware that variance of $d\mu$ is $\lVert \alpha \rVert^2$, but the parameter $c$ is sort of "scaled" so that the density function seems reduced to the normal one.
Could anyone please check for me?