Measuring every set from the outside

Mathematics · Calculus & Analysis · ages 22-23

Name ______________________ Date ____________

A particular cover has interval lengths 0.2, 0.3, and 0.4. What total length does this cover have?
Answer: ______________
In the definition of outer measure, what is taken over all interval covers?
- The infimum of the sums of their lengths
- The supremum of the sums of their lengths
- The number of intervals used in the cover
- The length of the single longest interval
What is the outer measure of the rational numbers inside [0, 1]?
- 1
- 0
- Infinite
What does the Caratheodory condition ask of a measurable set E?
- That E is closed, holding all of its boundary points
- That E can be covered by finitely many open intervals
- That E splits every test set into parts that add up
Why does a set with exactly three points have outer measure zero?
- A single point cannot be covered by any open interval
- A finite set is countable, so its outer measure is 0
- Three points always fit inside one interval of length 1
Every set of outer measure zero passes the Caratheodory test, so it is measurable. Is this true?
Circle one: True False
If E is contained in F, which inequality follows from covers?
- Outer measure of E is at most outer measure of F
- Outer measure of E is always larger
- The two outer measures must be equal
- Both outer measures must be infinite
Outer measure turns every countable union into an exact sum.
Circle one: True False
Which step in building a Vitali non-measurable set needs the axiom of choice?
- Listing all the rational numbers in one sequence
- Checking that every interval has its usual length
- Picking one member from each equivalence class
E1 = [0, 2] has outer measure 2 and E2 = [1, 4] has outer measure 3, so subadditivity bounds the outer measure of their union by 5. What is the actual outer measure of the union?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Measuring every set from the outside W1-mt_6u5_Um71wE-s1

0.9 · Add the three interval lengths: 0.2 plus 0.3 plus 0.4 equals 0.9.
The infimum of the sums of their lengths · Outer measure is the infimum of the total lengths of all open-interval covers.
0 · The rationals are countable, so tiny covers make their total length as small as you like.
That E splits every test set into parts that add up · A measurable set E satisfies the splitting equation for every test set A. It is not enough for the equation to hold for one selected test set.
A finite set is countable, so its outer measure is 0 · Finite sets are countable, and countably many points can be covered almost for free.
True · For any test set A, monotonicity makes the part inside E have outer measure 0 and the part outside at most the outer measure of A. Subadditivity gives the reverse bound, so the splitting equation holds for every A.
Outer measure of E is at most outer measure of F · Every cover of F also covers E, so taking infima gives outer measure of E at most outer measure of F.
False · Joined covers only give an inequality for unions. Equality can fail.
Picking one member from each equivalence class · No explicit rule picks a member from each of uncountably many groups, so the axiom of choice supplies the picks.
4 · The union is the interval [0, 4], whose outer measure is its length 4. That is below the bound 5 because the bound counts the overlap [1, 2] twice.