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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.MSC 2010: Primary: 28B05, 46G10, 46E10, 28C15. Secondary: 46B25, 46C05

We introduce and study an integral of Hilbert valued functions with respect to Hilbert valued measures. The integral is sesquilinear (bilinear in the real case) and takes scalar values. Basic properties of this integral are studied and some examples are introduced.

Based on the results from (Mihail and Miculescu in Math. Rep., Bucur. 11(61)(1): [21][22][23][24][25][26][27][28][29][30][31][32] 2009), where the shift space for an infinite iterated function system (IIFS for short) is defined and the relation between this space and the attractor of the IIFS is described, we give a sufficient condition on a family (I j ) j∈L of nonempty subsets of I, where S = (X, (f i ) i∈I ) is an IIFS, in order to have the equality j∈L A I j = A, where A means the attractor of S and A I j means the attractor of the sub-iterated function system S I j = (X, (f i ) i∈I j ) of S. In addition, we prove that given an arbitrary infinite cardinal number A, if the attractor of an IIFS S = (X, (f i ) i∈I ) is of type A (this means that there exists a dense subset of it having the cardinal less than or equal to A), where (X, d) is a complete metric space, then there exists S J = (X, (f i ) i∈J ) a sub-iterated function system of S, having the property that card(J) ≤ A, such that the attractors of S and S J coincide. MSC: Primary 28A80; secondary 54H25Keywords: infinite iterated function system (IIFS); sub-iterated function systems of a given IIFS; canonical projection from the shift space on the attractor of an IIFS; attractor of an IIFS

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