real analysis – Proof of Levy-Khintchine formula: Question on the existence of an infinitely divisible distribution defined by the formula

I am reading the proof of the Levy-Khintchine formula from Ken Iti Sato’s Levy Processes, however, I cannot understand a line from the proof that given a symmetric nonnegative definite $d times d$ matrix $A$, and a measure $nu$ on $mathbb{R}^d$ with $nu({0})=0$, $int (1 wedge |x|^2) nu(dx)<infty$, and $gamma in mathbb{R}^d$, we get an infinitely divisible distribution $mu$ whose characteristic function is given by the formula.

In the proof below, it states that $phi_n$ is the convolution of a Gaussian and a compound Poisson distribution. A distribution $mu$ on $mathbb{R}^d$ is compound Poisson if there exists $c>0$ and $sigma$ on $mathbb{R}^d$ with $sigma ({0})=0$ and the characteristic function $$hat mu(z)= exp(c(hat sigma(z)-1)).$$ $D$ here is the closed unit ball.

But I cannot see from below why we get this form for the inner integral. That is, it seems like we should have $int_{|x|>1/n} (1-ilangle z,xrangle 1_D(x)) nu(dx)=1$ from the definition above but I don’t see how we get this. Why does this integral define the characteristic function of a compound Poisson?

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