# Is there any relation of defined measure with Lebesgue measure?

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 $\mu$ ?

## 1 Answer

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).

• Sorry..when do you call measures to be equivalent? – Math Lover Sep 11 '16 at 16:13
• We call $\mu$ and $\lambda$ equivalent if $\mu$ is absolutely continuous with respect to $\lambda$ and vice versa. The definition of absolute continuity is under "Generalizations" here, if you haven't met it, but basically it just means that there's a function $f$ so that $\mu(A) = \int_A f d\lambda$ (by the Radon-Nikodym theorem). – Josh Keneda Sep 11 '16 at 16:17
• Thank you,i have not learned about the mentioned stuff yet.I'll have a look. – Math Lover Sep 11 '16 at 16:19