Uniqueness of finite Borel measures from their Fourier transforms

Set $\sigma =\mu -\nu$. It is a complex measure and its partition sums are bounded by $\mathrm{\mid }\mu \mathrm{\mid }+\mathrm{\mid }\nu \mathrm{\mid }$, so it has finite variation. Linearity of the bounded-test integrals gives $\widehat{\sigma }=0$. For every $t>0$, F1 gives a density $h_t\in L^1$ with zero transform; F2 gives $h_t=0$ almost everywhere. Passing to the F1 testing limit yields $\int \phi d\sigma =0$ for every complex $\phi \in C_c$.

Euclidean space is Hausdorff (disjoint small balls separate points) and locally compact by compact closed balls from F6. Balls with rational centers and positive rational radii form a countable base: for an open neighborhood of x choose a sufficiently small contained ball, then a rational center sufficiently near x and rational radius between the resulting strict bounds. Thus F3 applies to the finite positive measure $v=\mathrm{\mid }\sigma \mathrm{\mid }$ and makes it regular. Under AC, F5 gives Jordan parts $r^{+},r^{-}$ of $r=\mathrm{Re}\sigma$ and $s^{+},s^{-}$ of $s=\mathrm{Im}\sigma$. On their respective Hahn sets, for example $r^{+}(E)=r(E\cap P)\le \mathrm{\mid }\sigma (E\cap P)\mathrm{\mid }\le v(E)$; the same argument bounds each other part by v.

If $0\le \rho \le v$ is one of these parts and E is Borel, regularity of finite v gives compact $K\subseteq E$ and open $U\supseteq E$ with $v(E\setminus K)<ϵ$ and $v(U\setminus E)<ϵ$. The corresponding rho errors are at most these, so rho is both inner and outer regular and finite on compact sets, hence Radon. For real $\phi \in C_c$, step 1.1 gives $\int \phi d{r}^{+}=\int \phi d{r}^{-}$ and the analogous equality for s. These component integral identities follow for simple tests and then by their bounded-test approximation. F4 gives $r^{+}=r^{-}$ and $s^{+}=s^{-}$, so $\sigma =0$. AC is used in smoothing and the Hahn/Jordan decompositions, and covers the regularity construction.