theory. Since the late 20th century economists are having a renewed interest in the measurement issues of happiness. This field has been developing methods, surveys and indices to measure happiness. Several properties of Cardinal utility functions can be derived using tools from measure theory and set theory. Measurability A utility function is considered to be measurable, if the strength of preference or intensity of liking of a good or service is determined with precision by the use of some objective criteria. For example, suppose that eating an apple gives to a person exactly half the pleasure of that of eating an orange. This would be a measurable utility if and only if the test employed for its direct measurement is based on an objective criterion that could let any external observer repeat the results accurately. One hypothetical way to achieve this would be by the use of a hedonometer, which was the instrument suggested by Edgeworth to be capable of registering the height of pleasure experienced by people, diverging according to a law of errors. Before the 1930s, the measurability of utility functions was erroneously labeled as cardinality by economists. A different meaning of cardinality was used by economists who followed the formulation of Hicks-Allen. Under this usage, the cardinality of a utility function is simply the mathematical property of uniqueness up to a linear transformation. Around the end of the 1940s, some economists even rushed to argue that von Neumann-Morgenstern axiomatization of expected utility had resurrected measurability. The confusion between cardinality and measurability was not to be solved until the works of Armen Alchian, William Baumol, and John Chipman. The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature at the time. It is helpful to consider the same problem as it appears in the construction of scales of measurement in the natural sciences. In the case of temperature there are two degrees of freedom for its measurement - the choice of unit and the zero. Different temperature scales map its intensity in different ways. In the celsius scale the zero is chosen to be the point where water freezes, and likewise, in cardinal utility theory one would be tempted to think that the choice of zero would correspond to a good or service that brings exactly 0 utils. However this is not necessarily true. The mathematical index remains cardinal,