# Random measures on the Brownian path with prescribed expectation

Let B denote the range of the Brownian motion in R^d. For a deterministic Borel measure nu we wish to find a random measure mu such that the support of mu is contained in B and the expectation of mu is nu. We discuss when exactly can there be such a random measure and construct in those cases. We establish a formula for the expectation of the double integral with respect to mu, which is a strong tool for the geometric measure theory of the Brownian path.

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