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

‐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 3 is 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.

      There are many alternative ways of representing mathematically the unpredictable payout of this wager, as the following illustration shows. The outcome of the wager is the value , where

Examining each of the 36 pairs in (, ) in turn, the outcomes or payouts are listed in Table 2.1.

      For instance, of the 36 possible pairs of throws , a loss of 11 is incurred twice, with throws of (5,6) and (6,5).

      Accordingly, let the sample space for the wager be

      let the measurable sets consist of , the family of all subsets of , and, for let be as set out in Table 2.1; so, for , can be found by adding up the relevant probabilities in Table 2.1.

Table 2.1 Distribution of payouts.

Payout	Probability
1
2
3
4
5
‐11
‐12

      Now define random variable by the identity mapping for