For a random vector $X$ in $\Bbb{R}^d$, one can always define its characteristic function as the Fourier transform of the law $\mu_X$ of $X$:
$$ \varphi(\xi) = \Bbb{E}[\exp(i\xi^tx)] = \int_{\Bbb{R}^d}\exp(i\xi^tx)d\mu_X(x) $$ With various assumptions, one can recover $\mu_X$ from $\varphi$. For instance, if $\varphi\in L^1$, we know $\mu_X$ is absolutely continuous and we can apply the inverse Fourier transform to obtain its density. In one dimension, we have the following general formula (see e.g. Durrett Theorem 3.3.4 or Billingsly Theorem 26.2):
$$ \mu_X((a,b)) + \frac{1}{2}\mu_X(\{a,b\}) = \lim_{T\rightarrow\infty}\frac{1}{2\pi}\int_{-T}^T \frac{\exp(-ita) - \exp(-itb)}{it}\varphi(t)dt $$ An analogous formula should in theory hold in $d$ dimensions - does anyone have a reference? I imagine it would be more complicated because we might have $\mu_X$ concentrated on some $d-k$ dimensional subsets, but I would be interested in any related results.