Measure Theory/Countable Additivity

Properties of Length-measure

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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

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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

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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

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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

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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


To prove the upward continuity of measure, define the sequence of disjoint measurable sets,

 

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  .