  Let $B$ denote the range of the Brownian motion in $\mathbb{R}^{d}$
($d\geq3$). For a deterministic Borel measure $\nu$ on $\mathbb{R}^{d}$ we wish
to find a random measure $\mu$ such that the support of $\mu$ is contained in
$B$ and it is a solution to the equation $E(\mu(A))=\nu(A)$ for every Borel set
$A$. We discuss when it is possible to find a solution $\mu$ and in that case
we construct the solution. We study several properties of $\mu$ such as the
probability of $\mu\neq0$ and we establish a formula for the expectation of the
double integral with respect to $\mu\times\mu$. We calculate $\mu$ in terms of
the occupation measure when $\nu$ is the Lebesgue measure, i.e. we provide an
explicit deterministic density function of $\mu$ with respect to the occupation
measure. As a conclusion we calculate an explicit formula for the expectation
of the double integral with respect to the occupation measure. We generalise
the theory for more general random sets in separable, metric, Radon spaces. As
an additional example, we also apply our results to percolation limit sets on
boundaries of trees.
