This note contains characterizations of those sigma-fields for which sigma-finiteness is a necessary condition in the Radon-Nikodym Theorem.

Our purpose is to consider those σ-fields for which σ-finiteness is a necessary condition in the Radon–Nikodym Theorem. We first prove a measure theoretic equivalence in the general case, and then use this to obtain an algebraic characterization in the case when the σ-field is the Borel field of a locally compact separable metric space. For undefined terminology we refer the reader to [1] for measure theoretic and [2] for algebraic properties.

By a measure, we mean a countably additive function from σ-field of sets or a Boolean σ-algebra into the non-negative extended real numbers. We will say that a measure μ on a σ-field of sets Σ is RN provided each μ-continuous finite measure on Σ has a Radon–Nikodym derivative in L1(μ).

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π*s A particular example concerns non-singular bilinear maps ℝm+1 × ℝn+1 → ℝm+n+1−p; on restriction and normalisation these become biskew maps Sm × Sn ℝ Sm+n-p;. Now the Hopf construction ℋ applied to any map f: Sm × Sn → Sm+n-p yields

A function

analytic and univalent in U = {z: |z| < 1} is said to be starlike there, if f(U) is f starshaped with respect to the origin, that is, if w ε f(U) implies tw ε f(U) for 0 ≤t ≤ 1. We denote by S* the class of all such functions. The Koebe function; k(z) = z(l – z)-2, z ε U, maps U onto the complex plane minus a slit along the I negative real axis from - ¼ to ∞, and thus belongs to the class S*. Recently Leung [4] has shown that, if

then, for f ε S*,

for every p > 0.

Problems of classifying and enumerating types of plane patterns, tilings, and other repeating geometrical structures have interested mathematicians, crystallographers and others for many years. Recently we have formulated the general principles that seem to underlie many of the published treatments of these topics, and so have been able to put on a mathematical basis classification criteria often justified mainly on intuitive grounds. In other words, we can now decide whether or not two given patterns are of the same “type”, at one of a number of different possible levels of classification, without relying on some vaguely expressed distinction based on a “feeling” as to whether the objects in question should be regarded as of distinguishable kinds, or not.

In Section 1 below I describe two measures of the complexity of a binary relation. J The theorem says that these two measures never disagree very much. Both measures of complexity arose in connection with Saharon Shelah's notion [5] of a stable firstt order theory; Shelah showed in effect that one measure is finite, if, and only if, the other is finite too. This follows trivially from the theorem below. I confess my main aim was not to get the extra information which the theorem provides, but to eliminate Shelah's use of uncountable cardinals, which seemed strangely heavy machinery for proving a purely finitary result. Section 2 below explains the modeltheoretic setting.

The classification of conies in the real or complex projective plane under the action of the appropriate group is simple and well known. We consider here the more complicated question of classifying conies in the complex projective plane ℂℙ2 under the action of the real projective group PGL (3, ℝ).

Reid (1974) derived “first approximations” to solutions of the Orr–Sommerfeld equation, which are uniformly valid in a full neighbourhood of a critical point. This paper shows that such approximations may be calculated to higher order, and makes a first step towards placing the theory on a rigorous basis by providing error bounds for the dominant-recessive approximations. These are obtained by generalizing methods discussed by Olver (1974) for second order linear ordinary differential equations.