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lecture3

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

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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 Σ is a σ -algebra on a set S , and that μ : Σ [0 , ] is a measure. If A 1 , A 2 , ... are subsets in Σ satisfying A 1 A 2 ⊆ · · · then we have that if A = j A j , lim k →∞ μ ( A k ) = μ ( A ) . Note: We had proved this before for probability measures, but it holds for measures in general. MCT2. Suppose S, Σ , and μ are as above. Given A 1 A 2 A 3 · · · , all lying in Σ, let A be their intersection. Then, we have lim k →∞ μ ( A k ) = μ ( A ) . Note: This is a homework problem you were asked to solve, except that here μ is a general measure, not just a probability measure. Now, suppose that μ is the Lebesgue measure on B ( R ). Then, we have the following two basic observations: I. For any a R , μ ( { 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, μ ( { a } ) = lim j →∞ μ ( A j ) = lim j →∞ 2 j = 0 . 1

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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 λ is a measure on B ( R ) then λ ( { 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 λ ( A ) = 1 , if 0 A ; 0 , if 0 6∈ A. It is easy to check that this is a measure on B ( R ), and it is obvious that λ ( { 0 } ) = 1. The natrual progression of our discussion of the Lebesgue measure at this point would be to define the Lebesgue integral, and then state and prove various theorems about it. However, to do this task properly would take too
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lecture3 - Notes 3 for Honors Probability and Statistics...

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