Let (mu) be a finite measure on (left(mathbb{R}^{n}, mathscr{B}left(mathbb{R}^{n}ight)ight)) and set (chi(xi):=widehat{mu}(xi)). Show that (chi) is real-valued

Question:

Let $\mu$ be a finite measure on $\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)$ and set $\chi(\xi):=\widehat{\mu}(\xi)$. Show that $\chi$ is real-valued if, and only if, $\mu$ is symmetric w.r.t. the origin, i.e. $\widetilde{\mu}(B):=\mu(-B)=\mu(B)$.

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