Measure Theory/The Measurable Sets Form a Sigma-algebra

The Measurable Sets Form a Sigma-algebra

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In a previous exercise you showed that if   then  . This is called "closure under complements".

Here we will prove other useful closure properties, such as closure under unions and closure under intersections.


Exercise 1. Not All Unions


Suppose that   is closed under all unions. That is to say, if   is any collection of measurable sets, then

 

Show that, in that case,  .

Does this make you suspect that   is or is not closed under arbitrary unions? (We will answer this question more formally in the next lesson.)

Closure Under Union

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Exercise 1. Not All Unions hints to us that   is not closed under so-called "arbitrary unions". But perhaps there is some weaker closure property for unions that it still satisfies?

Let's try to prove closure under just pairwise unions.

Let   and we will try to show that  .

Let   and we need to show that   splits A cleanly. As usual, we only need to show one direction,

 

because the other direction is handled automatically by subadditivity.

Because this proof can get complicated, it may help to name all the relevant components of the set   in the following way.

  •  
  •  
  •  
  •  

Notice that with this naming system,  . Therefore the right-hand side of the inequality that we seek to prove is

 

which is now the same as

 


Exercise 2. Strategic Subadditivity


Complete the proof of closure under pairwise union by following these steps.

1. Strategically apply subadditivity.  That is to say, do not fully distribute the   to every unioned set above.  Keep inside of the   those sets with initial index 1.
2.  For the stuff left not distributed, reorganize this into  .
3.  Use the fact that F is measurable, applied to the set  , to "factor out" a  .  Then reorganize this.
4.  Use the fact that E is measurable.

Closure under Finite Unions

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Now that we have closure under pairwise unions, this generalizes easily to closure under finite unions.


Exercise 3. Formalize Closure under Finite Unions


This time I won't tell you exactly the theorem to state. Rather, it is your job to both formalize the theorem for finite unions, and then to prove it.

Closure under Countable Unions

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Finally we show closure under countable unions.

Let   be any countable collection of measurable sets, and  . As always, we need

 

By the result for finite additivity, we know that for each  ,

 


Exercise 4. Part of Countable Unions


Explain why  


Exercise 5. The Other Part of Countable Unions


Prove that   by completing the following steps.

1. For a fixed n, prove  .  Then simplify   and  .  
2. Use (1.) to prove by induction that  .  
3. Prove that  .
4. Take a limit as   and infer the result.

Intersections

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Exercise 6. Closed under Countable Intersection


Use closure under complements and closure under countable unions to easily infer closure under countable intersections.

Sigma Algebra

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Definition:  -algebra

Let X be any set (of real numbers, or anything else) and let  . If   satisfies the following three properties then we call   a  -algebra.

  •  
  • closure under complements
  • closure under countable unions

Note that the symbol  , when transliterated into English, is written "sigma".


Exercise 7. Measurable Sets Form a Sigma-algebra


Prove that   is a  -algebra. (The proof should merely reference results that we've already proved elsewhere.)

(Note: As of right now the concept of a  -algebra isn't terribly useful. However, if and when I extend this course to cover topics in general measure theory rather than just Lebesgue measure, the concept will become more important.)

All Intervals

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In a previous lesson we showed that the open rays,  , are all measurable.


Exercise 8. All Intervals Are Measurable


Now use the  -algebra properties of   to show that every other interval is measurable. Note: Intervals may be closed, open, or neither. They may also be bounded or unbounded.