A Simple Mathematical Framework for Learning and Teaching Probability Theory

Random variables with arbitrary distributions as well as large classes of stochastic processes can be constructed on \((0,1)^d\) with uniform distribution. Treating topics such as the law of large numbers or the central limit theorem entirely within this probability space, one can avoid to expose students to an unsound amount of general measure theory without giving up any mathematical rigor. In the light of an understanding of probability theory as a theory of typical physical behaviour, such as taught by Detlef Dürr, I argue that such a procedure entails no loss of generality. Observing the limitations of this procedure, one is led to general measure theory as a tool to overcome them.