  One considers Hilbert space valued measures on the Borel sets of a compact
metric space. A natural numerical valued integral of vector valued continuous
functions with respect to vector valued functions is defined. Using this
integral, different norms (we called them Monge-Kantorovich norm, modified
Monge-Kantorovich norm and Hanin norm) on the space of measures are introduced,
generalizing the theory of (weak) convergence for probability measures on
metric spaces. These norms introduce new (equivalent) metrics on the initial
compact metric space.
