The Lebesgue Integral and What It Repairs · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Slicing the range instead of the domain

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

  1. The indicator of the interval from 1 to 4 integrates to the interval's length. What is the integral?

    Answer: ______________

  2. A simple function equals 7 on the set {1, 2, 3} and 0 elsewhere on [0, 5]. What is its integral?

    • 21
    • 0
    • 7
    • 35

  3. How do you integrate a simple function?

    • Multiply its largest value by the whole domain's measure
    • Add value times measure over its pieces
    • Add its values without weighting them by any measure

  4. What is the Lebesgue integral over [0, 3] of the indicator of the rationals?

    • 0
    • 3
    • It has no integral

  5. A bounded function on [0, 1] is Riemann integrable. What follows for its Lebesgue integral?

    • It exists and equals the Riemann value
    • It exists but is always larger
    • It can never exist

  6. On [0, 2], the function f(x) = x has Lebesgue integral 2, the same as its Riemann integral. Is this true?

    Circle one:   True   False

  7. On [0, 2], what upper and lower Riemann sums does the rationals indicator have for every partition?

    • Upper sum 1, lower sum 0
    • Upper sum 2, lower sum 2
    • Upper sum 2, lower sum 0
    • Upper sum 0, lower sum 0

  8. Riemann sums for the rationals indicator stall because every subinterval holds both rational and irrational points.

    Circle one:   True   False

  9. A student says the rationals indicator should integrate to 1 since rationals sit everywhere. What is the flaw?

    • Being everywhere is not the same as having positive measure
    • The rationals cannot be listed, so no integral can be computed
    • Indicator functions of infinite sets can never be integrated

  10. On [0, 4], f equals 100 on the rationals. At irrational points it equals 2 on [0, 1], 5 on (1, 3] and 0 on (3, 4]. What is its Lebesgue integral?

    Answer: ______________

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Answer key

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

Slicing the range instead of the domain W1-mt_KO92nLc5YU-s1

  1. 3 · Length 4 minus 1 equals 3.
  2. 0 · A finite set has measure 0, so the piece with value 7 adds 7 times 0. The rest adds 0 times 5. The integral is 0.
  3. Add value times measure over its pieces · Each level contributes its height times the size of its piece.
  4. 0 · The rationals have measure zero, so only the zero level counts.
  5. It exists and equals the Riemann value · The new integral extends the old one without changing its values.
  6. True · f is continuous, so it is Riemann integrable with integral 2, the triangle area. On a bounded interval the Lebesgue integral agrees, so it is 2 as well.
  7. Upper sum 2, lower sum 0 · Every piece holds rationals and irrationals, so each supremum is 1 and each infimum is 0. The widths add to 2, so the upper sum is 2 and the lower sum is 0 for every partition.
  8. True · Lower sums stick at 0 while upper sums stick at 1.
  9. Being everywhere is not the same as having positive measure · Density differs from size: the rationals are dense but have measure zero.
  10. 12 · The rationals have measure 0, so the value 100 adds nothing. The irrational pieces add 2 times 1, 5 times 2 and 0 times 1, which is 12.
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