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

The set of outcomes of a single throw of a coin is the set , and the family of subsets of is

and is a measurable space. Define the following function to represent the coin tossing experiment:

Then, for , so is a ‐measurable function.

There are many different ways of defining the probability space. It is natural to use real‐number‐valued functions, so the outcomes can be denoted and instead of H and T, and the measurable function can then be the identity function. (It is usual that the values of a random variable are represented as real numbers 2 ; with expected—mean or average—value, variance, and so on; which are also real numbers.)

But no matter what way this construction is done, the classical, rigorous mathematical representation by measurable function is evidently more complicated than the naive or natural view of the coin tossing experiment. In contrast, the purpose of this book is to provide a rigorous theory of stochastic integration/summation which (like [MTRV]) bypasses the “measurable function” view, and which is closer to the “naive realistic” view.

Throw a pair of dice and, whenever the sum of the numbers observed exceeds 10, pay out a wager equal to the sum of the two numbers thrown, and otherwise receive a payment equal to the smaller of the two numbers observed. If the two are the same number (with sum not exceeding 10) then the payout is that number.

In Example 3take sample space

Observation of a throw of the pair of dice can be represented by a listing of the possible joint outcomes , , . Define a random variable by

for each . Then, as in the previous example where the domain and range of are finite sets, is ‐measurable and qualifies as a random variable, with expected value

The integral in this case reduces to the sum of a finite number of terms.

The payoff from the wager in Example 3is a randomly variable amount given by

In this case, is a composite of the deterministic function with the random variable ; and, just like , is (trivially) ‐measurable, and is a random variable, with

where, again, the Lebesgue integral reduces (trivially) to a finite sum of terms.