Mathematics 713 Assignment #6

    1.  Let Gn be any sequence of functions from R to R and G be
        defined by G(x) = inf{ Gn(x) : n=1,2,3, ... }.  Prove or
        disprove the claim that

             -1           ∞    -1
            G  ((a,∞)) =  ∩  Gn  ((a,∞)) ,
                         n=1

        in otherwords, prove or disprove that the inverse image of
        the set { y : y>a } under G is equal to the intersection
        of the inverse images under each Gn.

    2.  Let E be a subset of R.  Suppose for every ε > 0 there is 
        a Borel measurable set F such that F ⊆ E and 

              *
             λ (E\F) < ε .

        Prove or disprove the claim that E is Borel measurable.
  
        In otherwords, suppose the Lebesgue outer measure of E\F
        can be made arbitrarily small by choosing a suitable Borel
        set F which is also a subset of E.  Prove or disprove the
        claim that E is Borel measurable.

    3.  McDonald and Weiss problem 3.33

    4.  McDonald and Weiss problem 3.35

    5.  [Extra Credit]  McDonald and Weiss problem 3.39

    6.  Suppose E is Lebesgue measurable and λ(E) > 0.  Prove or
        disprove the claim that there is an open interval I such
        that λ(E∩I) > λ(I)/2.

        In otherwords, if Lebesgue measureable set E has positive
        Lebesgue measure, prove or disprove there is an interval 
        such that the Lebesgue measure of E intersected with that
        interval is more than half the length of the interval.