19.2 Measures And Their Properties
First we define what is meant by a measure.
Definition 19.2.1 Let
be a measurable space. Here ℱ is a σ algebra of sets of
] is called a measure if whenever
i=1∞ is a sequence of disjoint sets of ℱ, it follows
Note that the series could equal ∞. If μ
< ∞, then μ is called a finite measure. An important case is
when it is called a probability measure.
Note that μ
Example 19.2.2 You could have P
ℱ and you could define μ
to be the number of
elements of S. This is called counting measure. It is left as an exercise to show that this is a measure.
Example 19.2.3 Here is a pathological example. Let Ω be uncountable and ℱ will be those sets
which have the property that either the set is countable or its complement is countable. Let μ
if E is countable and μ
if E is uncountable. It is left as an exercise to show that this is a
Of course the most important measure is Lebesgue measure which gives the “volume” of a subset of ℝn.
However, this requires a lot more work.
Lemma 19.2.4 If μ is a measure and Fi ∈ℱ, then μ
. Also if Fn ∈ℱ and
Fn ⊆ Fn+1 for all n, then if F
If Fn ⊇ Fn+1 for all n, then if μ
< ∞ and F
= ∩nFn, then
Proof: Let G1 = F1 and if G1,
have been chosen disjoint, let
Thus the Gi are disjoint. In addition, these are all measurable sets. Now
and so μ
Now consider the increasing sequence of Fn ∈ℱ. If F ⊆ G and these are sets of ℱ
. If any
there is nothing to prove. Assume then that these
are all finite. Then
Next suppose μ
is a decreasing sequence. Then
is increasing to F1 ∖ F and so by the first part,
This is justified because μ
and all numbers are finite by assumption.