Let F be a family of subsets of X. It is well known that any measure m on X can be additively approximated by a measure m*, so that for every S in F:

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* | m(S) - m*(S) | < epsilon m(X)

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* the size of the support of m* is O( VCdim(F) / epsilon^2)

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We ask whether there exists a parameter of F analogous to VCdim, which governs the size of support of any m* that multiplicatively approximates m, i.e., for every S in F: (1-epsilon) m*(S) < m(S) < (1-epsilon) m*(S).

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It turns out that such a parameter exists. It is defined as the length of the longest sequence of sets in F, so that no S in this sequence is contained in the union of the sets preceding it. We call it trk(F), the triangular rank of F. We discuss some properties of trk(F), present another relevant parameter, and discuss some applications.

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Based on a joint work with Ilan Newman.