Convergence and relative compactness in distribution of sequences of random hypermeasures

Let Φ be a compact set in a vector space equipped with a convergence which is metrizable in Φ but not certainly in the whole space. We endow the space of continuous on Φ linear functionals on span Φ with the norm \( {\left\| u \right\|_\Phi } = \sup \varphi \in \Phi \left| {u\varphi } \right| \) and call the elements of the completion of Φ hypermeasures. We prove theorems on the convergence in probability or in distribution and relative compactness in distribution of a sequence of random hypermeasures.