Beurling and Deny [1], [2] introduced the notion of Dirichlet spaces. They [2] showed the existence of balayaged measures and equilibrium measures in the theory of Dirichlet spaces. In this paper, we shall show that the following equivalence is valid for a Dirichlet space on a locally compact Hausdorff space X.

Throughout this paper we assume that all rings contain an identity. We say that R is a semi-primary ring if its (Jacobson) radical N is nilpotent, and R/N is an Artinian ring. We say that R admits a splitting, and we write R=A+B if A is a. subring of R, if B is a two-sided ideal in R, and if A∩B=0 .

Recently A. Hattori introduced in [7], [8], [9] the notion of simple algebras over a commutative ring. Especially, in [9], he examined, as a fundamental problem on simple algebras, whether a directly indecomposable semi-simple algebra is simple or not and gave affirmative answers to this in some particular cases. In this note we shall first prove, as a complete answer to this, that any directly indecomposable semi-simple algebra over a Noetherian ring is simple.

Dans ce mémoire, on fera pour les fonctions algébroïdes des études analogues à ceux des fonctions méromorphes dans |z|<∞ faits dans Doob [3], Toda [7].

Soit g(z) une fonction méromorphe dans |Z|<∞ ayant un point singulier essentiel isolé à l’infini. J.L. Doob a obtenu le théorème suivant en utilisant la topologie fine pour g(z).

In this note, we treat the positive sentential logic LPS and the primitive sentential logic (the positive implicational calculus of Hilbert) LOS1). LOS has ‘implication’ as the only logical symbol and is a subsystem of LPS.

The conditions (TI 1) and (TI 2) are stated for and henceforth in the paper H is understood to be when D is taken to be a T.I. subset of G. Also in the definition of T. I. subset the condition is that D ∩ DG≠ø where ø is the empty set.