Abstract
In this chapter we shall derive the important inequalities for moduli of path families which form the basis for the geometric part of qr theory. The main reference for path families in connection with qc theory is the book [V4] by J. Vaisala. When convenient, we shall refer to [V4] rather than repeat proofs. The main results, namely Poletskiĭ’s and Vaisala ‘s inequalities, involve a somewhat technical measure theoretic step concerning path families whose modulus is neglible. We shall call this step Poletskiĭ’s lemma. In qc theory a corresponding result is known as Fuglede’s theorem. Inequalities for path families are more general and more effective than inequalities for capacities of condensers, which historically came first [MRV1]. Here we shall obtain the capacity inequalities as corollaries in Section 10.
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