Corollary

Let $\varrho$ be a unitary representation of $\mathbf{R}$ in a Hilbert space E. There exist a locally compact space X, a positive measure $\mu$ on X and a continuous function $g$ on X with real values such that $\varrho$ is isomorphic to the representation $\pi$ of $\mathbf{R}$ in $L^2(X, \mu)$ defined by $\pi (t)f=e^{itg}f$ for every $t\in \mathbf{R}$ and every $f\in L^2(X, \mu)$.

Let $u$ be the self-adjoint operator on E such that $\varrho (t) =e^{itu}$ for every $t\in \mathbf{R}$ (Theorem 1). There exist a locally compact space X, a positive measure $\mu$ on X, an isometric isomorphism $\theta$ of $L^2(X, \mu)$ onto E and a continuous function $g$ on X with real values such that $u=\theta \circ m_g\circ \theta^{-1}$ (Th. 1 of IV, p. 266). The assertion follows from the formula $e^{itu}=\theta \circ e^{itm_g}\circ \theta^{-1}=\theta \circ m_{e^{itg}}\circ \theta^{-1}$ (Lemma 4 of IV, p. 269).