Patrick Muldowney - Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

This formulation assumes that the aggregative operation , involving infinitely many random variables ( ), produces a single random entity whose expected value can be obtained by means of the operation .

Additionally, is said to be a Lebesgue integral‐type construction. The part of this statement should be unproblematical. The domain is a real interval, and has a distance or length function, which, in the context of Lebesgue integration on the domain, gives rise to Lebesgue measure on the space of Lebesgue measurable subsets of . So can also be expressed as .

However, the random variable‐valued integrand is less familiar in Lebesgue integration. Suppose, instead, that the integrand is a real‐number‐valued function . Then the Lebesgue integral , or , is defined if the integrand function f is Lebesgue measurable. So if J is an interval of real numbers in the range of f , the set is a member of the class of measurable sets; giving

That is, for each J , is a Lebesgue measurable subset of . This is valid if, for instance, f is a continuous function of s , or if f is the limit of a sequence of step functions.

How does this translate to a random variable‐valued integrand such as ? Two kinds of measurability arise here, because, in addition to being a ‐measurable function of , is a random variable (as is ), and is therefore a P ‐measurable function on the sample space :

Likewise . For to be meaningful as a Lebesgue‐type integral, the integrand must be ‐measurable (or ‐measurable) in some sense. At least, for purpose of measurability there needs to be some metric in the space of ‐measurable functions , , with , :