ABSTRACT
When dealing with statistical experiments whose outcomes form a continuous set of real numbers t, e.g., IR, we do not have to consider every subset of IR as an event, since every run of the experiment will give a value in a certain interval. The generally adopted subsets are Borel sets of the reals, named after the French mathematician Emile Borel (1871–1956). Borel sets, which include all open and closed intervals, individual numbers as well the empty set and IR, are large enough to be able to quantify all experimental outcomes. Before looking into set functions let us start by considering real-valued functions f defined on the real line IR. Borel functions are so pervasive that it is difficult to write down a function which is not Borel. Borel functions play an important role in the theory of Lebesgue integration since they are related to the integrability of functions.
