Density inversion from an integrable characteristic function

Put $C=(2\pi {)}^{-1}\int \mathrm{\mid }{\phi }_X\mathrm{\mid }<\mathrm{\infty }$. The defining integral is absolutely convergent and $\mathrm{\mid }f(x)\mathrm{\mid }\le C$. For $x_n\to x$, the integrands converge pointwise and are dominated by $\mathrm{\mid }{\phi }_X\mathrm{\mid }$, so DCT gives $f(x_n)\to f(x)$. Under the assumed AC the sequential criterion proves continuity. Conjugating the integral and substituting $s=-t$, conjugate symmetry yields $\overline{f(x)}=f(x)$; hence f is real.

By F1 and absolute convergence, for every $a<b$ $\mu ((a,b))+\frac{1}{2}\mu (\{a,b\})=\frac{1}{2\pi }\int q_{a,b}(t){\phi }_X(t)dt,q_{a,b}(t)={\int }_a^b{e}^{-ity}dy\mathrm{.}$ Its modulus is at most $C(b-a)$. Taking $a=x-1\mathrm{/}n,b=x+1\mathrm{/}n$, the point x lies in the open interval, so positivity and monotonicity give $\mu (\{x\})\le \mu ((a,b))\le C(b-a)=2C\mathrm{/}n$ for every n, hence every singleton has mass zero. Also the double absolute integral of $e^{-ity}{\phi }_X(t)$ on $(a,b)\times \mathbb{R}$ is $(b-a)\mathrm{\parallel }{\phi }_X{\mathrm{\parallel }}_1$, so Fubini gives $\mu ((a,b))={\int }_a^{b}f(y)dy$.

If $f(x_0)<0$, continuity supplies an interval about $x_0$ on which $f<f(x_0)\mathrm{/}2<0$, contradicting the nonnegative interval mass in step 2.1. Thus $f\ge 0$, and F4 defines a Borel measure $\nu (B)={\int }_{B}f$. Applying continuity from below to $(-n,n)\uparrow \mathbb{R}$ for both measures gives $\nu (\mathbb{R})=\mu (\mathbb{R})=1$. Bounded open intervals together with the empty set form a pi-system generating the Borel sets: their rational-endpoint subfamily is a countable base for the real topology. Step 2.1 and F5 therefore imply $\nu =\mu$. AC is inherited from F1 and covers the sequential continuity use in step 1.1.