851lecture05

851lecture05 - Statistics 851(Fall 2013 Prof Michael...

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Statistics 851 (Fall 2013) September 13, 2013 Prof. Michael Kozdron Lecture #5: The Borel Sets of R We will now begin investigating the second of the two claims made at the end of Lecture #3, namely that there exists a σ -algebra B 1 of subsets of [0 , 1] on which it is possible to define a uniform probability. Our goal for today will be to define the Borel sets of R . The actual construction of the uniform probability will be deferred for several lectures. Recall that a set E R is said to be open if for every x E there exists some > 0 (depending on x ) such that the interval ( x , x + ) is contained in E . Also recall that intervals of the form ( a, b ) for −∞ < a < b < are open sets. A set F R is said to be closed if F c is open. Note that both R and are simultaneously both open and closed sets. If we consider the collection O of all open sets of R , then it follows immediately that O is not a σ -algebra of subsets of R . (That is, if A O so that A is an open set, then, by definition, A c is closed and so A c / O .) However, we know that σ ( O ), the σ -algebra generated by O , exists and satisfies O σ ( O ) 2 R . This leads to the following definition. Definition. The Borel σ -algebra of R , written B , is the σ -algebra generated by the open sets. That is, if O denotes the collection of all open subsets of R , then B = σ ( O ).

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