[Inquiry] Re: Probability And Statistics

[Jon Awbrey]

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PAS.  Note 3

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| 1.2.  Probability Spaces (cont.)
|
| A nonempty collection of subsets of a given set !W! that is closed under
| finite set theoretic operations is called a 'field of subsets' of !W!.
| It therefore seems we should demand that $A$ be a field of subsets.
| It turns out, however, that for certain mathematical reasons just
| taking $A$ to be a field of subsets of !W! is insufficient.
| What we will actually demand of the collection $A$ is
| more stringent.  We will demand that $A$ be closed
| not only under finite set theoretic operations
| but under countably infinite set theoretic
| operations as well.  In other words if
| {A_n}, n >= 1, is a sequence of sets
| in $A$, we will demand that:
|
|    |_| (n = 1 to oo) A_n  is in  $A$
|
| and
|
|    |^| (n = 1 to oo) A_n  is in  $A$.
|
| Here we are using the shorthand notation:
|
|    |_| (i = 1 to oo) A_i  =  A_1 |_| A_2 |_| ...
|
| to denote the union of all the sets of the sequence, and:
|
|    |^| (i = 1 to oo) A_i  =  A_1 |^| A_2 |^| ...
|
| to denote the intersection of all the sets of the sequence.
|
| A collection of subsets of a given set !W! that is closed
| under countable set theory operations is called a !s!-field
| of subsets of !W!.  (The !s! [sigma] is put in to distinguish
| such a collection from a field of subsets.)  More formally we
| have the following:
|
| Definition 1.
|
| A nonempty collection of subsets $A$ of a set !W!
| is called a !s!-field of subsets of !W! provided
| that the following two properties hold:
|
|    1.  If A is in $A$, then A^c is also in $A$.
|
|    2.  If A_n is in $A$, n = 1, 2, ..., then:
|
|           |_| (i = 1 to oo) A_i
|
|        and
|
|           |^| (i = 1 to oo) A_i
|
|        are both in $A$.
|
| Hoel, Port, Stone, 'Probability Theory', p. 7.
|
| Hoel, P.G., Port, S.C., & Stone, C.J.,
|'Introduction to Probability Theory',
| Houghton Mifflin, Boston, MA, 1971.

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