There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributions.

Given two probability distributions expressing returns on two single risky assets of a portfolio, we innovatively define two consumer’s demand functions connected with two contingent consumption plans. This thing is possible whenever we coherently summarize every probability distribution being chosen by the consumer. Since prevision choices are consumption choices being made by the consumer inside of a metric space, we show that prevision choices can be studied by means of the standard economic model of consumer behavior. Such a model implies that we consider all coherent previsions of a joint distribution. They are decomposed inside of a metric space. Such a space coincides with the consumer’s consumption space. In this paper, we do not consider a joint distribution only. It follows that we innovatively define a stand-alone and double risky asset. Different summary measures of it characterizing consumption choices being made by the consumer can then be studied inside of a linear space over $${\mathbb {R}}$$ R . We show that it is possible to obtain different summary measures of probability distributions by using two different quadratic metrics. In this paper, our results are based on a particular approach to the origin of the variability of probability distributions. We realize that it is not standardized, but it always depends on the state of information and knowledge of the consumer.

If we study the expected utility function then we deal with a unified approach to an integrated formulation of decision theory in its two subjective components: utility and probability. We decompose the expected utility function inside of an m-dimensional linear space after decomposing a contingent consumption plan viewed as a univariate random quantity. We propose a condition of coherence compatible with all possible attitudes in the face of risk of a consumer. It is a geometric condition of coherence. In particular, we consider a risk-neutral consumer and his coherent decisions under uncertainty. The right closed structure in order to deal with utility and probability is a linear space in which we study coherent decisions under uncertainty having as their goal the maximization of the prevision of the utility associated with a contingent consumption bundle.

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This paper focuses on logical aspects of choices being made by the consumer under conditions of uncertainty or certainty. Such logical aspects are found out to be the same. Choices being made by the consumer that should maximize her subjective utility are decisions studied by revealed preference theory. A finite number of possible alternatives is considered. They are mutually exclusive propositions identifying all quantitative states of nature of a consumption plan. Each proposition of it is expressed by a real number. This research work distinguishes it from its temporary truth value depending on the state of information and knowledge of the consumer. Since each point of the consumption space of the consumer belongs to a two-dimensional convex set, this article focuses on conjoint distributions of mass. Indeed, the consumption space of the consumer is generated by all coherent summaries of a conjoint distribution of mass. Each point of her consumption space is connected with a weighted average of states of nature of two consumption plans jointly studied. They give rise to a conjoint distribution of mass. The consumer chooses a point of a two-dimensional convex set representing that bundle of goods actually demanded by her inside of her consumption space. This paper innovatively shows that it is nothing but a bilinear and disaggregate measure. It is decomposed into two real numbers, where each real number is a linear measure. In this paper, different measures are obtained. They can be disaggregate or aggregate measures, where the latter are independent of the notion of ordered pair of consumption plans.