Is there any relation of defined measure with Lebesgue measure?

Question

Let $M(n,\mathbb R)$ denotes the $n \times n$ matrices,by identifying $M(n,\mathbb R)$ with $\mathbb R^{n^2}$ we have a measure on $M(n,\mathbb R)$ namely the $n^2$ dimensional Lebesgue measure.In a set of notes the following measure is defined: For any subset $A \subset M(n, \mathbb R)$ consider $$\mu(A)=\frac {1}{(2\pi)^{k/2}} \int_A e^\frac {-\sum x_i^2}{2} d\lambda(X)$$ where $\lambda$ denote the Lebesgue measure.Is there any relation between $\lambda$ and $\mu$ ? Is there any name for measure $\mu$ ?

Answer 1

The name for $\mu$ is the standard Gaussian measure. See here for some basic traits. It's especially worth noting that $\mu$ is a probability measure and it's equivalent to $\lambda$ (in the sense of absolute continuity).