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Radically Elementary Probability Theory. (AM-117)

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Nelson develops a new approach to probability theory
that is just as powerful as but much simpler than conventional
"Kolmogorov-style" probability theory used throughout mathematics
for most of the 20th century. This book has two "radical" innovations.

The first innovation is a very simple version of nonstandard analysis,
much simpler than Abraham Robinson's original version, developed in only eight
pages, just powerful enough to define the arithmetic of infinitesimal
and unlimited real numbers.

The second innovation is a very simple version of probability theory
that restricts all sample spaces to be finite (though perhaps unlimited)
and all probabilities of outcomes to be nonzero (though perhaps infinitesimal).
Thus there is no need for measure theory or any other PhD level mathematics.

As Nelson says in his preface, "the mathematical background required is little
more than that which is taught in high school, and it is my hope that it will
make deep results from the modern theory of stochastic processes readily
available to anyone who can add, multiply, and reason." The level
of sophistication required of readers is that of undergraduate math majors.
The reduction of technical difficulty can be seen by his going from nothing to
"a version of the de Moivre-Laplace central limit theorem that
contains Lindeberg's theorem on the sufficiency of his condition, Feller's
theorem on its necessity, Wiener's theorem on the continuity of the
trajectories of his process, the Levy-Doob characterization if it as the
only normalized martingale with continuous trajectories, and the invariance
principle of Erdos and Kac as extended by Donsker and Prohorov" in just 79
pages.

This book will be of no use to anyone who just wants to learn conventional
probability theory. But it is essential for anyone who wants a
perspective from outside conventional theory. Anyone brainwashed by
a PhD level probability course has a hard time seeing that probability
theory can be any other way (although it was very different in the 19th
century and many different alternative approaches were tried in the 20th).
Nelson's book is an eye opener.