Overview and State-of-the-Art of Uncertainty Visualization - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 8

principle could be known," but in practice is not. This type of uncertainty is called
epistemic and is subjective, such as not knowing the birth date of the last Chinese
Emperor. These uncertainties are due to errors that practically cannot be controlled
and can be described by non-probabilistic modeling.

1.3.2 Mathematical Modeling of Uncertainty

A variety of types of uncertainties occur in practice, including mixtures of different
types of uncertainty. Quantification of uncertainties, including mixtures, requires a
unifying mathematical framework, which is very difficult to establish and not yet
fully accomplished.

1.3.2.1 Fundamental Setting

From a fundamental standpoint, we are interested in the situation with possible
outcomes or occurrences of “events” A, B, C, where A, B, and C are subsets of
the set of all elementary events in the universe. The task at hand is to then measure
the evidence that A ever happened, the degree of truth of that statement “event A
happened”, and the probability that event A will happen. The question is then, how
do we measure and what is measurement?
In mathematics, measurement means to assign real numbers to sets. For example,
the classical task in metric geometry is to assign numbers to geometric objects for
length, area, or volume. The requirement in the measurement task is that the assigned
numbers should be invariant under displacement of the respective objects.
In ancient times, the act of measuring was equivalent to comparing with a standard
unit. However, it soon became apparent that measurement was more complicated
than initially thought in that it involves finite processes and sets. The first tool to
deal with this problem was the Riemann integral which enabled the computation of
length, areas, and volumes for complex shapes (as well as other measures). How-
ever, the Riemann integral has a number of deficiencies, including its applicability
only to functions with a finite number of discontinuities, fundamental operations of
differentiation and integration are, in general, not reversible, and limit processes, in
general, can not be interchanged. In 1898, Émile Borel developed classical measure
theory which includes -algebra to define a class of sets that is closed under set

union of countably many sets and set complement, and defined as additive measure
μ

σ

∈ R 0 with each bounded subset in the -algebra. Around

1899-1902, Henry Lebesgue defined integrals based on a measure that subsumes the
Borel measure, based on a special case. He connected measures of sets and measures
of functions.

that associates a number σ