Gnedenko and Kolmogorov [3, pp. 181-182] have shown that if Xn with law F(x) belong to the domain of normal attraction of a stable law of index 0<α<2, i.e. if partial sum Sn/an1/α converges in distribution to some stable law Vα, a>0 then there exist c1 and c2 such that

and

Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.

Despite the abundance of generalizations of Carathéodory's theorem occurring in the literature (see [1]), the following simple generalization involving infinite convex combinations seems to have passed unnoticed. Boldface letters denote points of Rn and Greek letters denote scalars.

The purpose of this note is to prove the following:

Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.

Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite if

for every finitely nonzero sequence {fn} of vectors in H [1].

In [4] Hatcher introduced the notion of an identity in an arbitrary category and proved a characterization of quasivarietal subcategories which is similar to Birkhoff's characterization of varietal subcategories in universal algebra. The aim of this note is to show that the theorem of Hatcher as well as the categorical generalization of Birkhoff's theorem are special cases of a "relative" theorem, formulated with respect to a projective structure.

It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

Let A be a k-dimensional Euclidean region having unit volume. An m-colors pattern is a partition PA of A into m sets Ai, i—1,…, m with positive volume. PA is called a random pattern if in addition the partition of A is a realization of a random process with the following stationarity and isotropy properties:

• (i) for all points s ∊ A, P(s ∊ Ai) =pi9, i = 1,..., m

• (ii) for all pair of points s, s'∊ A with distance |s—s|=d between them, P(s' ∊ Ai | s ∊ Aj)Pij(d) i,j= 1,..., m.

P. Erdös asked the following problem: Does there exist an infinite sequence of integers a1<…satisfying for every x≥1

1

so that every integer is of the form 2k+ai [1]. The analogous questions can easily be answered affirmatively if the powers of 2 are replaced by the rth power.

In this note we give a simple affirmative answer to the problem of Erdôs. Let c2 be a sufficiently small absolute constant. Our sequence A consists of all the integers of the form

2