7.3 Hyperanalysis of Probability

Here we will develop the hyperanalysis for probability theory. Since probability is a measure with values in [0,1], measure theory and Lebesgue integration theory need to be developed using hyperanalysis. But we will not pay too much attention to unbounded measures.

First notice that hyper measure spaces ^[*] are not ?-additive, except when the space has only finitely many elements. However, Loeb discovered that, as a consequence of the saturation property, such a space always extends uniquely to a ?-additive space. For convenience, we concentrate on the case when the measure is a probability measure, although the method works for any Borel measure. A special case is when the hyper measure space is obtained from the counting measure of a hyperfinite set. The Lebesgue measure on the real line and the Wiener measure can be obtained from such hyperfinite counting measure. In these cases, the standard integration corresponds to the internal hyperfinite summation. (Hence one deals with Riemann sum instead of Lebesgue integral.) One direction of the correspondence is the standard part of a function, the other direction is the lifting of a standard function to an S-integrable function.

For a Daniell style approach to Loeb integration, see [Hurd and Loeb, 1985] for details.