Math 540-Real Analysis- Homework 1
Due date: September 9
(1) Let R2be the sets of finite disjoint union of rectangles of the form
(a, b]×(c, d]
in the plane. Show that R2is an algebra. Show that every triangle belongs
to the σ-algebra generated by R2. Show that
m((a, b]×(c, d]) = (b−a)(d−c)
is σ-additive on R2.
(2) Problem 37-page (Show that the Cantor set obtained form iteratively re-
moving a middle interval of length 1/3, starting from [0,1] is a compact set.
Show also that every point xin the Cantor set can be written as
x=X
j
aj3−j
with aj∈ {0,2}. What about uniqueness?)
46
(3) Problem 38-page 46. (Show that there is a bijection between the Cantor
set and [0,1]-and review bijection!)
1