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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ).
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern