Q: Inverse Fourier formula for lattice random variables Suppose that $X$ has a lattice distribution such that no strict sublattice of $\mathbb{Z}^d$ contains $X$ with probability $1$. Define the characteristic function of $X$ as usual:
$$
\gamma(t)\equiv E\exp(it'X)=\sum_{x\in\mathbb{Z}^d}\exp(it'x)\Pr(X=x).
$$
My book is giving the following as the inversion formula for $\Pr(X=x)$:
$$
\Pr(X=x)=(2\pi)^{-d}\int_{[-\pi,\pi]^d}\exp(-it'x)\gamma(t)dt.
$$
Why is $t$ restricted to $[-\pi,\pi]^d$ please?
