Proposition 7.3.4: Properties of Outer Measure

Outer measure has the following properties:
  1. Outer measure m* is a non-negative set function whose domain is P(R), i.e. the power set of R.
  2. The outer measure of an interval is its length.
  3. Outer measure is countably subadditive, i.e. if { An } is a countable collection of sets, then
    m*( An) m*(An)
Context Context

Proof

The first statement is obvious. Outer measure is clearly defined for every set, therefore its domain is P(R). It is also non-negative because all terms involved in the inf are non-negative.

We have already shown that the outer measure of open and closed intervals is their length, so we do not have to prove the second statement again.

Only the third statement requires a proof - which will be given later.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007