Outer Measure and the Construction of Lebesgue Measure · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Measuring every set from the outside

Mathematics · Calculus & Analysis · ages 22-23
Name ______________________   Date ____________

  1. Outer measure of an interval equals its length. Interval from 1 to 4 has what outer measure?

    Answer: ______________

  2. What is the outer measure of the rationals in [1, 2]?

    • 1
    • 0.5
    • 0
    • Infinite

  3. What is the outer measure of the rational numbers inside [0, 1]?

    • 1
    • 0
    • Infinite

  4. What does the Caratheodory condition ask of a measurable set E?

    • That E holds all of its boundary points
    • That E can be covered by finitely many intervals
    • That E splits every test set into parts that add up

  5. Why does a set with exactly three points have outer measure zero?

    • Points are too small to cover at all
    • Any finite set is countable, and countable sets get zero
    • Three points always sit inside one interval

  6. Vitali construction uses the axiom of choice to pick representatives, giving a nonmeasurable set. Is this correct?

    Circle one:   True   False

  7. Which set passes the Caratheodory test and is measurable?

    • Closed interval [1, 4]
    • Vitali nonmeasurable set
    • Any nonmeasurable set
    • Set splitting some test set nonadditively

  8. Outer measure turns every countable union into an exact sum.

    Circle one:   True   False

  9. Which step in building a Vitali non-measurable set needs the axiom of choice?

    • Listing all the rational numbers
    • Checking that intervals have lengths
    • Picking one member from each shifted group

  10. Outer measures are 2 and 3. Subadditivity bounds the union by the sum. What is the bound?

    Answer: ______________

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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

  1. 3 · Length 4 minus 1 equals 3.
  2. 0 · 0 is correct since countable sets have outer measure zero, while 1, 0.5, and infinite overcount.
  3. 0 · The rationals are countable, so tiny covers make their total length as small as you like.
  4. That E splits every test set into parts that add up · E is measurable exactly when each test set satisfies the splitting equation.
  5. Any finite set is countable, and countable sets get zero · Finite sets are countable, and countably many points can be covered almost for free.
  6. True · Choice picks one per rational equivalence class, and the resulting set cannot be measurable.
  7. Closed interval [1, 4] · Closed interval [1, 4] is measurable, while Vitali and other nonmeasurable examples fail additivity.
  8. False · Joined covers only give an inequality for unions. Equality can fail.
  9. Picking one member from each shifted group · No rule names a pick from each of uncountably many groups, so choice does that work.
  10. 5 · 2 plus 3 equals 5.
Worksheet · LightMySky