Math 410 Homework, due Thursday March 24
(1) (a) Find an example of a measure zero subset A of Rn whose
closure A does not have measure zero in Rn . (b) Find an example of a measure zero subset B of Rn whose boundary Bd(B)
does not have measure zero in Rn .
(2) Show that no open subset of Rn has measure zero in Rn .
(3) Show that the set Rn−1 × {0} has measure zero in Rn .
(4) Show that the set of irrational numbers in [0, 1] does not have
measure zero in R.
(5) Let Q be a rectangle in Rn and let f : Q → R be a bounded
function. Show
that if f vanishes on a closed set B of measure
R
zero, then Q f exists and equals zero.
(6) Show that if Q1 , Q2 , . . . is a countable collection P
of rectangles
n
in R which cover the rectangle Q, then vol(Q) ≤ ∞
i=1 vol(Qi ).
(These problems are all from Chapter 3 of Munkres’s book “Analysis
on Manifolds.”)
1