  We prove the following variant of Marstrand's theorem about projections of
cartesian products of sets:
  Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$
respectively, and $\pi:\mathbb R^{m_1}\times...\times\mathbb R^{m_n}\to\mathbb
R^k$ be a surjective linear map. We set $$\mathfrak{m}:=\min\{\sum_{i\in
I}\dim_H(K_i) + \dim\pi(\bigoplus_{i\in I^c}\mathbb R^{m_i}),
I\subset\{1,...,n\}, I\ne\emptyset\}.$$ Consider the space $\Lambda_m=\{(t,O),
t\in\mathbb R, O\in SO(m)\}$ with the natural measure and set
$\Lambda=\Lambda_{m_1}\times...\times\Lambda_{m_n}$. For every
$\lambda=(t_1,O_1,...,t_n,O_n)\in\Lambda$ and every $x=(x^1,,x^n)\in\mathbb
R^{m_1}\times...\times\mathbb R^{m_n}$ we define
$\pi_\lambda(x)=\pi(t_1O_1x^1,...,t_nO_nx^n)$. Then we have $(i)$ If
$\mathfrak{m}>k$, then $\pi_\lambda(K_1\times...\times K_n)$ has positive
$k$-dimensional Lebesgue measure for almost every $\lambda\in\Lambda$. $(ii)$
If $\mathfrak{m}\leq k$ and $\dim_H(K_1\times...\times
K_n)=\dim_H(K_1)+...+\dim_H(K_n)$, then \dim_H(\pi_\lambda(K_1\times...\times
K_n))=\mathfrak{m}$ for almost every $\lambda\in\Lambda$.
