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

. Taking , let be the set consisting of the 16 elements

      and let be an atomic measure with if , and, for ,

      (2.16)

for or ; with for all other . Thus, if Up and Down transitions are independent and equally likely, then and

      (2.17)

      As in previous versions of the sample space , this construction imposes probabilities on the outcomes or , by means of the relation

      for measurable sets in the range of , since is clearly a measurable function on the domain . Thus, for outcomes or , and

      

2.5 More on Basic Stochastic Integral

      The constructions in Sections 2.3 and 2.4 purported to be about stochastic integration. While a case can be made that (2.6) and (2.7) are actually stochastic integrals, such simple examples are not really what the standard or classical theory of Chapter 1 is all about. The examples and illustrations in Sections 2.3 and 2.4 may not really be much help in coming to grips with the standard theory of stochastic integrals outlined in Chapter 1.

      This is because Chapter 1, on the definition and meaning of classical stochastic integration, involves subtle passages to a limit, whereas (2.6) and (2.7) involve only finite sums and some elementary probability calculations.

      From the latter point of view, introducing probability measure spaces and random‐variables‐as‐measurable‐functions seems to be an unnecessary complication. So, from such a straightforward starting point, why does the theory become so challenging and “messy”, as portrayed in Chapter 1

As in Example 2, the illustration in Section 2.3 involves dividing up the time period (4 days) into 4 sections; leading to sample space