ThmDex – An index of mathematical definitions, results, and conjectures.
Fourier transform characterises euclidean real probability measure
Formulation 0
Let $M = (\mathbb{R}^D, \mathcal{B}(\mathbb{R}^D))$ be a D2763: Euclidean real Borel measurable space such that
(i) $\mu, \nu : \mathcal{B}(\mathbb{R}^D) \to [0, 1]$ are each a D198: Probability measure on $M$
(ii) $\mathfrak{F}_{\mu}$ and $\mathfrak{F}_{\nu}$ are each the D4131: Finite unsigned euclidean real Borel measure Fourier transform of $\mu$ and $\nu$, respectively
Then \begin{equation} \mathfrak{F}_{\mu} = \mathfrak{F}_{\nu} \quad \iff \quad \mu = \nu \end{equation}
Proofs
Proof 0
Let $M = (\mathbb{R}^D, \mathcal{B}(\mathbb{R}^D))$ be a D2763: Euclidean real Borel measurable space such that
(i) $\mu, \nu : \mathcal{B}(\mathbb{R}^D) \to [0, 1]$ are each a D198: Probability measure on $M$
(ii) $\mathfrak{F}_{\mu}$ and $\mathfrak{F}_{\nu}$ are each the D4131: Finite unsigned euclidean real Borel measure Fourier transform of $\mu$ and $\nu$, respectively