A Gaussian-weighted measure — absolutely continuous both ways

Let R denote the real line, B the Borel $\sigma$-field and $m$ the Lebesgue measure on R. For $A\in\mathbf B$, define $\mu(A)=\int_A\exp(-x^2/2)\,dm(x)$. Consider the statements: (I) $\mu$ is a measure on B. (II) The Radon–Nikodym derivative of $\mu$ with respect to $m$, $\dfrac{d\mu}{dm}(x)=\exp(-x^2/2)$ a.e. $m$. (III) $m<<\mu$ ($m$ is absolutely continuous with respect to $\mu$).