Measurability invariance, continuity and a portfolio representation

Galileo suggested that what is not measurable be made measurable. It is this principle which underscores an unwritten law of both the sciences and the social sciences that it is better to measure than not to measure. But, the assumption of measurability is rarely considered. In this paper, we consider a set of invariance and continuity conditions which a measure should satisfy. These conditions provide a test of whether a given mapping onto the real line constitutes a measure, and not simply an arbitrary mapping. They represent a test for measurability. In the social sciences, it is common to construct measures based on multi-dimensional attributes. In the paper, we characterise this multi-dimensional measurement as portfolios, with weights determined a priori. Measurement becomes a process of convergence towards a preferred measure which anchors the measurement. Measurement is valid if there is convergence to a measure satisfying the invariance and continuity conditions.

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