p3 - j = 1 2 the set E n,j:= x ∈ R 1 g n x< 1/j is...

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #3 (due on W, October 13). 50 points are divided between 5 problems, 10 points each. #1. Let f be a real function on R 1 . The image and the inverse image of a subset A R 1 under f are correspondingly f ( A ) = { y : y = f ( x ) for some x A } , f - 1 ( A ) = { x : f ( x ) A } . Show that f ( f - 1 ( A )) A f - 1 ( f ( A )) for arbitrary A R 1 . Give an example when f - 1 ( f ( A )) 6 = A . Hint. You can use without proof the properties of f - 1 on p.4, and Proposition 0.23 on p.14. #2. Let f ( x ) be a continuous function on R 1 , and let B denote the Borel σ -algebra on R 1 . Show that M := f - 1 ( B ) := { E : E = f - 1 ( F ) for some F ∈ B} is a σ -algebra contained in B . #3. Let f ( x ) be an arbitrary bounded function on R 1 . Show that the set E = { x R 1 : lim y x f ( y ) = f ( x ) } is Borel measurable. Hint. Consider the sequence of functions g n ( x ) := sup I n,k f - inf I n,k f for x I n,k := ± k - 1 2 n , k 2 n i ; k = 0 , ± 1 , ± 2 , . . . . For each x 6 = k/ 2 n , we have x E g n ( x ) 0 as n → ∞ . Then note that for each
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Unformatted text preview: j = 1 , 2 , . . . , the set E n,j := { x ∈ R 1 : g n ( x ) < 1 /j } is Borel measurable (why?) #4. Let A be an arbitrary nonempty set, and for each α ∈ A , let an interval I α := [ a α , b α ] be defined with 0 ≤ a α < b α ≤ 1. Show that there is a finite or countable subset A ⊆ A , such that [ α ∈ A I α = [ α ∈ A I α . Hint. Start with a similar property for intervals I α containing a fixed rational point r k ∈ [0 , 1]. #5. The Cantor set (see p.38) C := n x = ∞ X j =1 a j 3 j , where a j = 0 or 2 for all j o . Find the algebraic sum C + C := { z = x + y : x, y ∈ C } . Hint. First find 1 2 · C + 1 2 · C ....
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