{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis 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 de±ne a uniform probability. Our goal for today will be to de±ne the Borel sets of R .T h ea c t u a lc o n s t r u c t i o no ft h e 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 )suchtha tthein te rva l( x °, x + ° )i scon ta inedin 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 ,thenitfo l lowsimmed iate lythat O is not a σ -algebra of subsets of R .(Tha ti s ,i f A ∈O so that A is an open set, then, by de±nition, A c is closed and so A c / .) However, we know that σ ( O ), the σ -algebra generated by O , exists and satis±es O ° σ ( O ) 2 R .Th i sl ead stothefo l low ingde±n i t ion . DeFnition. The Borel σ -algebra of R ,w r i t ten B sthe σ -algebra generated by the open sets. That is, if O denotes the collection of all open subsets of R ,then B = σ ( O ).
Background 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 ]}

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online