2 Characteristic functions
The characteristic function of a measure \(\mu \) on a normed space \(E\) is the function \(E^* \to \mathbb {C}\) defined by
\begin{align*} \hat{\mu }(L) = \int _E e^{i L(x)} \: d\mu (x) \: . \end{align*}
In a separable Banach space, if two finite measures have same characteristic function, they are equal.
Proof
The characteristic function of a measure \(\mu \) on an inner product space \(E\) is the function \(E \to \mathbb {C}\) defined by
\begin{align*} \hat{\mu }(t) = \int _E e^{i \langle t, x \rangle } \: d\mu (x) \: . \end{align*}
This is equal to the normed space version of the characteristic function applied to the linear map \(x \mapsto \langle t, x \rangle \).
In a separable Hilbert space, if two finite measures have same characteristic function, they are equal.
Proof
Let \(\mu \) be a measure on \(F\) and let \(L \in F^*\). Then
\begin{align*} \widehat{L_*\mu }(x) & = \hat{\mu }(x \cdot L) \: . \end{align*}
Proof