Sigma-Algebras and Measurable Sets · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Which sets get a size

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

  1. Universe has elements 1, 2, 3. Smallest sigma algebra containing {1} has how many sets?

    Answer: ______________

  2. Universe {1, 2}. Which collection is a sigma algebra?

    • Empty set and {1} only
    • Empty set, whole set, {1}, {2}
    • {1} and {2} only
    • {1} only

  3. Which of these must belong to every sigma-algebra on a space X?

    • Every set with exactly one point
    • Every finite set
    • The whole space X

  4. A sigma-algebra is closed under complements: with every set it also holds that set's complement.

    Circle one:   True   False

  5. How do you get that a sigma-algebra is closed under countable intersections?

    • Intersect the sets two at a time forever
    • Complement each set, unite them, complement back
    • List the sets in increasing size order

  6. Collection with only the empty set and the whole set is always a sigma algebra. Is this correct?

    Circle one:   True   False

  7. How is the Borel sigma algebra described?

    • Set of all open intervals only
    • Set of all finite sets only
    • Power set of rationals only
    • Smallest sigma algebra containing all open intervals

  8. On X = {1, 2}, is the family with the empty set, {1}, and X a sigma-algebra?

    • No, because the complement of {1} is missing
    • Yes, because it holds unions and complements
    • No, because it holds too many sets

  9. A classmate says closing under finite unions is enough for limit arguments. Which reply spots the error?

    • Limit sets can need countably many steps, so finite rules fall short
    • Finite unions are never allowed inside a sigma-algebra
    • A countable union is only a finite union written twice

  10. Universe {a, b}. How many distinct sigma algebras exist?

    Answer: ______________

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Answer key

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

Which sets get a size W1-mt_cpvegazufk-s1

  1. 4 · Required sets are empty, {1}, {2, 3}, whole set, totaling 4.
  2. Empty set, whole set, {1}, {2} · Empty set, whole set, {1}, {2} is the full power set, closed under complements and unions, while each other collection misses the whole set or complements.
  3. The whole space X · A sigma-algebra always contains the whole space. Single points and finite sets need not belong.
  4. True · That is one of the two closing rules.
  5. Complement each set, unite them, complement back · The De Morgan laws turn an intersection into complements plus a union.
  6. True · Trivial collection is closed under complements and unions.
  7. Smallest sigma algebra containing all open intervals · Smallest sigma algebra containing all open intervals is the definition, while opens alone lack complements and the other options are too narrow.
  8. No, because the complement of {1} is missing · The complement of {1} is {2}, which is not in the family.
  9. Limit sets can need countably many steps, so finite rules fall short · Sets built by limits, like points in infinitely many sets, need countable unions.
  10. 2 · Only trivial with 2 sets and discrete with 4 sets work, so total is 2.
Worksheet · LightMySky