Measure Theory/Countable Additivity
Properties of Length-measure
[edit | edit source]We can finally achieve what has been elusive for so long: a measure of sets of real numbers, which is countably additive.
In this lesson, we prove that the which we just constructed is countably additive.
Theorem: Countable Additivity
is countably additive. |
Pairwise Additive
[edit | edit source]As a warmup to countable additivity, let's prove the easier claim of pairwise additivity. Let be two measurable sets which are disjoint, .
We would like to show that .
Exercise 1. Apply Measurability
Prove the desired result by applying the measurability of F to . Don't forget to use the fact that E and F are disjoint. |
Exercise 2. Generalize
Notice that the proof did not actually require E to also be measurable. Therefore state a generalization of the above result. |
Exercise 3. Finite Additivity
Prove by induction that is therefore finitely additive. As part of the exercise, state what "finitely additive" should mean. |
Exercise 4. Countable Additivity
Let be any countable collection of disjoint measurable sets. We would like to show To do so, state the result which you just proved for finite additivity. Then apply monotonicity and then take the limit as . Finally, use the above to prove countable additivity. |
Exercise 5. Use Countable Additivity
Find and then find . Also find . |
Excision
[edit | edit source]We have additivity results, and one would hope that we have something like results which resemble subtraction.
Definition: excision
Let be any two measurable sets and . Then the property that is called excision.
Exercise 6. Prove Excision.
Prove that satisfies excision. |
Continuity of Measure
[edit | edit source]Recall that, roughly stated, if a function f is continuous then . Effectively, continuity of f means that the limit passes into the function.
There is a similar property for length measure. If is an ascending sequence of measurable sets (i.e. for ) then
One small problem with the statement above is that the expression is ... not even defined, actually.
But of course it makes good sense to identify this as .
(Also note that is superfluous because the sequence is ascending. This is just the same thing as .)
Definition: continuity of measure
Let be an ascending sequence of measurable sets. The property that
is called the upward continuity of measure.
A sequence of sets is called "descending" if for .
Definition: continuity of measure
Let be a descending sequence of measurable sets, and assume has finite measure. The property that
is called downward continuity of measure.
Proof
[edit | edit source]We now set for ourselves the proof of the theorem.
Theorem: Continuity of Measure
satisfies both upward and downward continuity of measure. |
Exercise 7. Prove the Upward Continuity of Measure
Show that . Next apply countable additivity. Finally, justify and then use the fact that for each . |
Exercise 8. Prove the Downward Continuity of Measure
To prove the downward continuity of measure, let be as in the statement of the definition. Define the ascending sequence of sets . 1. Prove that this sequence is ascending, and then apply the upward continuity of measure. 2. Infer downward continuity of measure. |
Exercise 9. Counterexample
Give an example descending countable sequence of measurable sets, , such that . |