The maximum sum of sizes of non-empty cross $t$-intersecting families

Let $[n]:=\{1,2,\dots ,n\}$, and $M$ be a set of positive integers. Denote the family of all subsets of $[n]$ with sizes in $M$ by $\left(\genfrac{}{}{0px}{}{\left[n\right]}{M}\right)$. The non-empty families $\mathcal{A}\subseteq \left(\genfrac{}{}{0px}{}{\left[n\right]}{R}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0px}{}{\left[n\right]}{S}\right)$ are said to be cross $t$-intersecting if $\mathrm{\mid }A\cap B\mathrm{\mid }\ge t$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. In this paper, we determine the maximum sum of sizes of non-empty cross $t$-intersecting families, and characterize the extremal families. Similar result for finite vector spaces is also proved.