Applications of the Extension Theories

The most important consequences of the extension theories of chapter II appear to be the representation theorems of Daniell-Stone and Riesz type which will be obtained in chapter V. The present chapter is devoted to a few direct applications of the extension theories to different areas in the domain of set functions. All these are central areas of rich tradition and extensive literature. We do not intend to compete with them in technical depth and sophistication. We rather want to show that certain basic ideas and results come as natural outflows from the above theories. We shall arrive at certain forms which are more comprehensive and simpler, and also more explicit than those found in the literature so far. An example is the extension of the capacitability theorem due to Choquet in section 10. In most but not all cases we shall be concerned with the conventional outer and inner situations.