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

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

rel="nofollow" href="#fb3_img_img_b22e9934-9ef0-5c57-8bf7-5010c5f45536.png" alt="normal upper Omega equals StartSet negative 5 comma negative 4 comma negative 3 comma negative 2 comma negative 1 comma 0 comma 1 comma 2 comma 3 comma 4 comma 5 comma 10 EndSet"/>

whose elements are the different values which can be taken by the variable w left-parenthesis 4 right-parenthesis . For instance, upper A equals StartSet negative 1 comma 0 EndSet , which is a member of the family script upper A of all subsets of normal upper Omega .

Following through the logic of the classical theory, probability upper P is defined on the family script upper A of measurable subsets of normal upper Omega . A random variable script upper W is a real‐number‐valued, and left-parenthesis upper P comma script upper A right-parenthesis ‐measurable, function

script upper W 4 colon normal upper Omega right-arrow from bar bold upper R comma script upper W 4 left-parenthesis omega right-parenthesis right-arrow w left-parenthesis 4 right-parenthesis

in this case, where the potential values w left-parenthesis 4 right-parenthesis element-of bold upper R are the numbers in the right‐most column of Table 2.4. The latter set is finite; and every finite subset, such as StartSet negative 1 comma 0 EndSet , is measurable. In fact, with sample space normal upper Omega chosen in this way, script upper W 4 is the identity function, since we have chosen so that its elements are the distinct values w left-parenthesis 4 right-parenthesis .

To find the probability of a set upper S of w left-parenthesis 4 right-parenthesis ‐outcomes, such as upper S equals StartSet negative 1 comma 0 EndSet , the classical theory requires that the corresponding set script upper W 4 Superscript negative 1 Baseline left-parenthesis upper A right-parenthesis element-of script upper A be found so that

upper P left-parenthesis upper S right-parenthesis equals upper P left-parenthesis script upper W 4 Superscript negative 1 Baseline left-parenthesis upper A right-parenthesis right-parenthesis

gives the probability of outcomes as the corresponding probability in the sample space. Conveniently, in this case normal upper Omega is chosen as simply the set of outcomes left-brace w left-parenthesis 4 right-parenthesis right-brace ; script upper W 4 is the identity function; and

upper S equals upper A equals script upper W 4 Superscript negative 1 Baseline left-parenthesis upper A right-parenthesis comma so upper P left-parenthesis upper S right-parenthesis equals upper P left-parenthesis script upper W 4 Superscript negative 1 Baseline left-parenthesis upper A right-parenthesis right-parenthesis

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 w left-parenthesis 4 right-parenthesis , and are not primarily inherited from some abstract measurable space normal upper Omega .

Alternatively, let the sample space be bold upper R and let script upper A be the class of Borel subsets of (so includes the singletons StartSet omega EndSet for each omega element-of normal upper Omega ). Define upper P on by upper P left-parenthesis bold upper R minus normal upper Omega right-parenthesis equals 0 and

upper P left-parenthesis StartSet omega EndSet right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column one sixteenth 2nd Column if omega equals negative 5 comma 0 comma 1 comma 2 comma 3 comma 4 comma 5 comma or 10 comma 2nd Row 1st Column two sixteenths 2nd Column if omega equals negative 1 comma negative 2 comma negative 3 comma or negative 4 comma 3rd Row 1st Column 0 2nd Column otherwise semicolon EndLayout

so upper P is atomic. As before, with upper S equals StartSet negative 1 comma 0 EndSet ,

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 normal upper Omega , in which probability measure upper P is specified, is often chosen—as above—in such a way that measure‐theoretic abstractions and complexities melt away, so that the “natural” or untutored approach, involving just outcomes and their probabilities, is applicable.

[MTRV] shows how to formulate an effective theory of probability

Скачать книгу