Quasi-measures and dimension theory

The theory of quasi-measures on compact Hausdorff spaces has been initiated by J. Aarnes. Here the relation of quasi-measures to classical notions of dimension theory is studied in the more general setting of normal spaces. It is shown that if X is normal and the large inductive dimension Ind(X) ⩽ 1, then every quasi-measure on X admits a unique extension to a closed regular, finitely additive measure on the Borel algebra of X. As a consequence, every quasi-linear functional on Cb(X), the space of bounded continuous real valued functions on X, is linear.