target="_blank" rel="nofollow" href="#fb3_img_img_98b94b33-e07c-50f2-abaa-fc24ec16c7dc.png" alt="upper S equals StartSet negative 1 comma 0 EndSet"/>, the classical theory requires that the corresponding set gives the probability of outcomes as the corresponding probability in the sample space. Conveniently, in this case
trivially. In effect, the random‐variable‐as‐measurable‐function approach of classical theory reduces to the “naive” or “realistic” method, in which the probabilities pertain to outcomes
Alternatively, let the sample space be
so
Classical probability involves a quite heavy burden of sophisticated and complicated measure theory. There are good historical reasons for this, and it is unwise to gloss over it. In practice, however, the sample space
[MTRV] shows how to formulate an effective theory of probability which follows naturally from the naive or realistic approach described above, and which does not require the theory of measure as its foundation. The following pages are intended to convey the basic ideas of this approach.
Before moving on to this, here is an elaboration of a technical point of a financial character, which appeared in Example 5 above and in the ensuing discussion, and which is relevant in stochastic integration.
Example 6
Expression (2.5) above gives two representations of a stochastic integral,
based on sample value calculations (2.4:
(2.10)
If
