Probabilistic Metric Spaces by B. Schweizer A. Sklar

Author:B. Schweizer,A. Sklar
Language: eng
Format: epub
Publisher: INscribe Digital

whence it follows that there is no triangle function under which (S, ) is a proper PM space.

10.2. Consistency, Triangle Inequalities

Let Cpq, Cqr and Cpr be the subcopulas associated with the d.f.’s defined by (10.1.8) and (10.1.9). Then

These subcopulas are incompatible (see Section 6.6). For if there were a 3-copula Cpqr such that

and

then we would have , whence (6.1.2) would yield

a contradiction. It is this incompatibility that lies at the root of the counterexample of the preceding section.

Theorem 10.2.1. Let (S, ) be a distribution-generated space over Rn. For any p, q in S let Cpq be a 2n-copula of Gp Gq, and Hpq, so that

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