Let (mu) be a finite measure on (left(mathbb{R}^{n}, mathscr{B}left(mathbb{R}^{n}ight)ight)). Prove P. Lvy's truncation inequality: [muleft(mathbb{R}^{n} backslash[-2 R,

Let \(\mu\) be a finite measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\). Prove P. Lévy's truncation inequality:

\[\mu\left(\mathbb{R}^{n} \backslash[-2 R, 2 R]^{n}ight) \leqslant 2\left(\frac{R}{2}ight)^{n} \int_{[-1 / R, 1 / R]^{n}}\left(\mu\left(\mathbb{R}^{n}ight)-\operatorname{Re} \breve{\mu}(\xi)ight) d \xi\]

[show that the right-hand side equals

\[2 \int\left(1-\prod_{1}^{n} \frac{\sin \left(x_{n} / Right)}{x_{n} / R}ight) \mu(d x)\]

shrink the integration domain to \(\mathbb{R}^{n} \backslash[-2 R, 2 R]^{n}\) and observe that \(0 \leqslant \frac{1}{2} \sin 2 \leqslant \frac{1}{2}\) and \(x^{-1} \sin x<1, x eq 0\).]