Mathematics 713 Assignment #5

    1.  Let Bn(f) be defined as in the proof of the Weierstrass
        approximation theorem in the handout.  Namely,

                          n            n!     k      n-k
            (Bn(f))(x) =  ∑  f(k/n) -------- x  (1-x)    .
                         k=0        k!(n-k)!

        Prove that (Bn(f)(x))^2 ≤ Bn(f^2)(x) for x in [0,1].

    2.  Let U and V be open subsets of R.  Prove or disprove the
        claim that
                          _   _   ___
                          U ∩ V = U∩V ,

        in otherwords, prove or disprove the intersection of the
        closures is equal to the closure of the intersection for
        open subsets of R.

    3.  Let E and F be collections of subsets of R.  Let A(E) be
        the sigma algebra generated by E, A(F) the sigma algebra
        generated by F and A(E∩F) the sigma algebra generated by
        E∩F.  Prove or disprove A(E)∩A(F)=A(E∩F), in otherwords,
        prove or disprove the sigma algebra of the intersection
        is equal to the intersection of the sigma algebras.

    4.  [Extra Credit]  Let X be an uncountable set and P be the
        power set P = { A : A ⊆ X }.  Prove or disprove P ~ X, 
        in otherwords, prove or disprove the claim there must be
        a bijection between an uncountable set and its power set.

    5.  McDonald and Weiss problem 3.12

    6.  McDonald and Weiss problem 3.14