lecture3 - Notes 3 for Honors Probability and Statistics...

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Unformatted text preview: Notes 3 for Honors Probability and Statistics Ernie Croot September 12, 2003 1 Measure Theory and Integration Now that we have a measure on B ( R ), namely the Borel-Lebesgue measure, we can try to see what measure it assigns to certain sets. The main tool we will use is the Monotone Convergence Theorem for Sets in one of the following two forms: MCT1. Suppose that is a -algebra on a set S , and that : [0 , ] is a measure. If A 1 , A 2 , ... are subsets in satisfying A 1 A 2 then we have that if A = j A j , lim k ( A k ) = ( A ) . Note: We had proved this before for probability measures, but it holds for measures in general. MCT2. Suppose S, , and are as above. Given A 1 A 2 A 3 , all lying in , let A be their intersection. Then, we have lim k ( A k ) = ( A ) . Note: This is a homework problem you were asked to solve, except that here is a general measure, not just a probability measure. Now, suppose that is the Lebesgue measure on B ( R ). Then, we have the following two basic observations: I. For any a R , ( { a } ) = 0. To see this, we apply MCT2: Let A j be the open interval ( a- 1 /j, a + 1 /j ). Then, { a } = j A j . So, ( { a } ) = lim j ( A j ) = lim j 2 j = . 1 II. As a corollary of I we get that any countable subset of R has measure 0 (why?). Here is a question that leads to a deeper appreciation of just how special the Lebesgue measure is: Question: Must we have that if is a measure on B ( R ) then ( { a } ) = 0? It turns out that the answer is NO; that is, there do exist strange measures on B ( R ) which assign singletons non-zero measures. A good example of such a measure is the following one: Suppose that A B ( R ). Then we have that ( A ) = 1 , if 0 A ; , if 0 6 A....
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This note was uploaded on 10/23/2011 for the course MATH 3225 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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lecture3 - Notes 3 for Honors Probability and Statistics...

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