Examples 7.3.9(b): Properties of Measure |
What is the measure of the set Q of all rational numbers and the
set I of all irrational numbers inside [0, 1].
Context
|
We have already shown that a countable set
has outer measure zero, and that sets with outer
measure zero are measurable. Therefore Q
is measurable with m(Q) = 0.
The set [0, 1] is an interval, hence it is measurable with
m([0, 1]) = 1 - 0 = 1. Also, the set
I = [0, 1] - Q, so that I is
also measurable. Since Q and I are disjoint, we can use
additivity of measure:
1 = m( [0, 1] ) = m(Q 
I) =
= m(Q) + m(I) = 0 + m(I)
Therefore m(I) = 1.
Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007