ABSTRACT
We begin by proving a pointwise relationship between the upper and lower limits as r → 0 of µ(B(x, r))/m(B(x, r)), and those as t → 0+ of u(x, t), where µ is a non-negative measure on R̲n, u is its Gauss-Weier-strass integral, and x is fixed. This motivates a study of the former limits, in which we establish that, if µ is singular with respect to Lebesgue measure, then µ(B(x, r))/m(B(x, r)) → ∞ as r → 0, for µ-almost all x ∊ R̲n. This involves a sophisticated covering theorem, which we shall use again in Chapter VII. Our property of singular measures is then used to prove some special representation theorems for non-negative temperatures. These results then interact with the notion of a thermic majorant of the positive part of a temperature, to produce other representation theorems and a variant of the generalized maximum principle, both for temperatures which are not necessarily non-negative.
