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

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 , :

      For example, the “distance” between and could be

      With such a metric at hand, it may then be possible to define , or , as the limit of the integrals of (integrable) step functions converging to for , as .

      Unfortunately, most standard textbooks do not give this point much attention. But for relatively straightforward integrands such as , it should not be too difficult.

      Continuing the discussion of I1, I2, I3, I4, it appears that the output of this definition of stochastic integral is a random entity ; perhaps a process which is some collection of random variables .

      Again comparing this with basic integration of a real number‐valued function , the integral is some kind of average or weighted aggregate value for . This integral, if it exists, produces a single unique real number (depending on the value of t), denoted by .

      For random variable‐valued integrand , suppose (for the purpose of speculation) that the stochastic integral

(if it exists) is equivalent (in some unspecified sense) to a single, unique random variable . Remember, a random variable is a function, usually real‐valued⁵, defined on a sample space . Two such functions, and