  We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that
there exists a random subset $\Theta$ of $\mathbb{S}^{d-1}$ such that the limit
of the derivative martingale exists simultaneously for all directions $\theta
\in \Theta$ almost surely. This allows us to define a random measure on
$\mathbb{S}^{d-1}$ whose density is given by the derivative martingale.
  The proof is based on first moment arguments: we approximate the martingale
of interest by a series of processes, which do not take into account the
particles that travelled too far away. We show that these new processes are
uniformly integrable martingales whose limits can be made to converge to the
limit of the original martingale.
