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In this section we discuss some topics from measure theory that are a bit more advanced than the topics in the previous sections of this chapter. However, measure-theoretic ideas are essential for a deep understanding of probability, since probability is itself a measure. In particular, σ-algebras play a fundamental role, even for applied probability, in encoding the state of information about a random experiment.
Suppose that is a set, playing the role of a universal set for a particular mathematical model. It is sometimes impossible to include all subsets of in our model, particularly when is uncountable. In a sense, the more sets that we include, the harder it is to have consistent theories. However, we almost always want the collection of admissible subsets to be closed under the basic set operations. This leads to some important definitions.
Suppose that is a collection of subsets of . Then is said to be an algebra (or field) if
Suppose that is an algebra of subsets of . Show that .
Suppose that is an algebra of subsets of and that for each in a finite index set . Show that
Thus it follows that an algebra of sets is closed under complements and under finite unions and intersections. However in many mathematical theories, probability in particular, this is not sufficient; we often need the collection of admissible subsets to be closed under countable unions and intersections.
Suppose that is a collection of subsets of . Then is said to be a σ-algebra (or σ-field) if
Clearly a σ-algebra of subsets is also an algebra of subsets, so the basic results for algebras still hold.
Show that if for each in a countable index set , then . Hint: Use DeMorgan's law.
Thus a σ-algebra of subsets of is closed under countable unions and intersections. This is the reason for the symbol σ in the name.
Suppose that is a set and that is a finite algebra of subsets of . Show that is also a σ-algebra. Hint: any countable union of sets in reduces to a finite union.
However, there are algebras that are not σ-algebras:
Show that the collection of co-finite subsets defined below is an algebra of subsets of , but not a σ-algebra:
Recall that denotes the collection of all subsets of , called the power set of . Trivially, is a the largest σ-algebra of , and as noted above, is sometimes too large to be useful. At the other extreme, the smallest σ-algebra of is given in the following exercise.
Show that is a σ-algebra.
In many cases, we want to construct a σ-algebra that contains certain basic sets. The following exercises show how to do this.
Suppose that is a σ-algebras of subsets of for each in a nonempty index set . Show that the intersection of the σ-algebras is also a σ-algebra of subsets of :
Suppose now that is a collection of subsets of . Think of the sets in as basic sets; but in general will not be a σ-algebra. The σ-algebra generated by is the intersection of all σ-algebras that contain , which by the previous exercise really is a σ-algebra:
Show that is the smallest σ algebra containing
Suppose that is a subset of . Show that
Suppose that and are subsets of . List the 16 (in general distinct) sets in . Open the Venn diagram applet and check your answer.
Suppose that is a collection of subsets of . Show that there are (in general distinct) sets in the σ-algebra generated by the given collection.
Suppose that is a set with σ-algebra , and that . Show that is a σ-algebra of subsets of . When (which is usually the case), note that . is the σ-algebra on induced by .
In this subsection, we will discuss some natural σ-algebras that are used for special types of sets. First, if is countable, we use the power set as the σ-algebra. Thus, all sets are admissible. For , the set of real numbers, we use the σ-algebra generated by the collection of all intervals. This is sometimes called the Borel σ-algebra, named after Emil Borel.
Suppose that is a sequence of sets and that is a σ-algebra of subsets of for each . For the product set
we use the σ-algebra generated by the collection of all product sets:
We extend this idea to an infinite product. Thus, suppose that is an infinite sequence of sets and that is a σ-algebra of subsets of for each . For the product set
we use the σ-algebra generated by the collection of all cylinder sets:
Combining the product construction with our earlier remarks about , note that for , we use the σ-algebra generated by the collection of all products of intervals. This is the Borel σ-algebra for .
Recall that a set usually comes with a σ-algebra of admissible subsets. Thus, suppose that and are sets with σ-algebras and , respectively. If , then a natural requirement is that the inverse image of any admissible subset of be an admissible subset of . Formally is said to be measurable if
Suppose that , , are sets with σ-algebras , , and , respectively. Show that if is measurable and is measurable, then is measurable.
Suppose that , and that is a σ-algebra of subsets of . Show that the collection
is a σ-algebra of subsets of , called the σ-algebra generated by . Hint: Recall that the inverse image preserves all set operations.
The σ-algebra generated by is the smallest σ-algebra on that makes measurable (relative to the given σ-algebra on ). More generally, suppose that is a set with σ algebra for each in a nonempty index set , and that for each . The σ-algebra generated by this collection of functions is
Again, this is the smallest σ-algebra on that makes measurable for each .
Most of the sets encountered in applied probability are either countable, or subsets of for some , or more generally, subsets of a product of a countable number of sets of these types. In this subsection, we will explore some of theses special cases.
Suppose that is countable and is given the σ-algebra . Show that any function on is measurable.
Recall that the set of real numbers is given the σ-algebra generated by the collection of intervals, the Borel σ-algebra. All of the real-valued elementary functions are measurable. The elementary functions include algebraic functions (which in turn include the polynomial and rational functions), the usual transcendental functions (exponential, logarithm, trigonometric), and the usual functions constructed from these.
Suppose that is a sequence of sets and that is a σ-algebra of subsets of for each . Recall that for the product set we use the σ-algebra generated by the collection of all product sets of the form where for each .
If is a function from into , then , where is the coordinate function, mapping into . As we might expect, is measurable if and only if is measurable for each .
We won't be overly pedantic about measure-theoretic assumptions in this project. Unless we say otherwise, we assume that all sets that appear are measurable (that is, members of the appropriate σ-algebras), and that all functions are measurable (relative to the appropriate σ-algebras).