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hmwk10sol_a - Math 202A Homework 10 Roman Vaisberg Problem...

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Math 202A Homework 10 Roman Vaisberg Problem 5. Suppose E is a given set, and O n is the open set: O n = { x | d ( x, E ) < 1 /n } . (a) Show if E is compact then, m ( E ) = lim n →∞ m ( O n ). Proof. Clearly we have O k O k +1 and E = n =1 O n . Thus O n E . Thus we would have the desired conclusion if we could show that m ( O 1 ) < . However, since E is compact subset of R d it must be bounded and thus it must be contained inside some cube, Q N of side length N centered at the origin. But then O 1 Q N +2 and since Q N +2 has finite measure, namely ( N + 2) d , we conclude that measure of O ! is finite. (b) Show the conclusion in (a) may be false for E closed and unbounded; or E open and bounded. Proof. First let E = { x R | x Z } . Then E is closed and unbounded. However, it is clear that for each n , m ( O n ) = . However E has measure 0 since it is countable. Now let ˆ C be a generalized Cantor set with positive measure. Let E = [0 , 1] - ˆ C . Then m ( E ) < 1. However since E is dense in [0 , 1] we have for each n , O n = ( - 1 n , 1+ 1 n ) and clearly lim n →∞ m ( O n ) = 1. Problem 7. If δ = ( δ 1 , . . . , δ d ) is a d -tuple of positive numbers δ i > 0, and E is a subset of R d , we define δE by δE = { ( δ 1 x 1 , . . . , δ d x d ) | where ( x 1 , . . . , x d ) E } . Prove that δE is measurable whenever E is measurable and m ( δE ) = δ 1 · · · δ d m ( E ) . Proof. To see that δE is measurable it suffices to observe that δ represents a linear transformation and Problem 8 gives the desired conclusion. Now let > 0. Since E is measurable we know for some collection { Q j } of closed cubes E j Q j and j m ( Q j ) < m ( E ) + δ 1 ··· δ d . Then we have δE j δQ j and thus m ( δE ) j m ( δQ j ). Since each δQ j
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