Measure Theory/The Measurable Sets Form a Sigma-algebra
The Measurable Sets Form a Sigma-algebra
[edit | edit source]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
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
[edit | edit source]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
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
[edit | edit source]Now that we have closure under pairwise unions, this generalizes easily to closure under finite unions.
Exercise 3. Formalize Closure under Finite Unions
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Closure under Countable Unions
[edit | edit source]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
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Exercise 5. The Other Part of Countable Unions
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.
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Intersections
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Exercise 6. Closed under Countable Intersection
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Sigma Algebra
[edit | edit source]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
(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
[edit | edit source]In a previous lesson we showed that the open rays, , are all measurable.
Exercise 8. All Intervals Are Measurable
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