  Given integers $r\geq 2$ and $n,t\geq 1$ we call families
$\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross
$t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have
$\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of
the classic Hilton-Milner theorem on cross intersecting families. In
particular, we determine the maximum of $\sum_{j\in
[r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the
cases when these are $k$-uniform families or arbitrary subfamilies of
$\mathscr{P}([n])$. Only some special cases of these results had been proved
before. We obtain the aforementioned theorems as instances of a more general
result that considers measures of $r$-cross $t$-intersecting families. This
also provides the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for
families of possibly mixed uniformities $k_1,\ldots,k_r$.
