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. On X = {1, 2, 3}, how many sets are in the smallest sigma-algebra containing {1}?

    Answer: ______________

  2. Which collection is a sigma-algebra on every space X?

    • The empty set and X
    • All one-point sets
    • All finite sets
    • All proper subsets of X

  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 show 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. The empty set belongs to every sigma-algebra. Is this true?

    Circle one:   True   False

  7. What separates an algebra of sets from a sigma-algebra?

    • An algebra need not be closed under taking complements
    • An algebra is closed under countable intersections, not unions
    • An algebra may lack closure under countable unions
    • An algebra may omit the whole space and the empty set

  8. On X = {1, 2, 3}, is the family with the empty set, {1}, {2}, 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. How many sigma-algebras exist on a two-element set {a, b}?

    Answer: ______________

LightMySky · lightmysky.comW1-mt_cpvegazufk-s1

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 · It contains the empty set, {1}, its complement {2, 3}, and X, for four sets.
  2. The empty set and X · The collection containing only the empty set and the whole space is closed under complements and countable unions for every X.
  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 · A sigma-algebra contains the whole space, and the complement of the whole space is the empty set.
  7. An algebra may lack closure under countable unions · An algebra is closed under complements and finite unions. A sigma-algebra also has closure under countable unions.
  8. No, because the complement of {1} is missing · The complement of {1} is {2, 3}, 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 · There are the trivial sigma-algebra {empty set, {a, b}} and the full power set, for a total of two.
Worksheet · LightMySky