Lemma 1.1.2 ([Lemma 8.1.1, Str24]).label For each $\phi \in \sps^{*}$, there exists a unique $\real^{d}$-valued Borel measure $\lambda$ on $[0, \infty)$ such that
- (1)
$\lambda(0) = 0$.
- (2)
$\angles{\theta, \phi}_{\sps}= \int_{[0, \infty)}\theta d\lambda$ for all $\theta \in \sps$.
- (3)
$\norm{\phi}_{\sps^*}= \int_{[0, \infty )}(1 + t) d\abs{\lambda}(t) < \infty$.
If a finite $\real^{d}$-valued Borel measure $\lambda$ on $[0, \infty)$ satisfies the above, then $\lambda$ defines a linear functional on $\sps^{*}$.
Proof. Let $\Lambda: \sps \to C_{0}((0, \infty); \real^{d})$ be the isomorphism defined above, then $\phi \circ \Lambda^{-1}\in C_{0}((0, \infty); \real^{d})^{*}$. By the Riesz representation theorem, there exists a vector measure $\lambda_{0} \in M((0, \infty); \real^{d})$ such that
and $\norm{\lambda_0}_{\text{var}}= \normn{\phi \circ \lambda^{-1}}_{\sps}$. Thus for any $\theta \in \sps$,
and
is the desired measure.$\square$