Measurable Functions and Approximation by Simple Functions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Functions that respect measurable sets

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

  1. Simple function equals 1 on the interval from 1 to 4 and 0 elsewhere. Its integral is 1 times length. What is it?

    Answer: ______________

  2. How to test measurability with rays?

    • Check the function is continuous
    • Check the function is bounded
    • Check the domain is finite
    • Check preimage of every open ray is measurable

  3. To test whether a function f is measurable, what do you check?

    • That f is continuous at every point
    • That f takes only finitely many values
    • That preimages of open rays are measurable sets

  4. A pointwise limit of measurable functions is always measurable.

    Circle one:   True   False

  5. Why do the powers x to the n on [0, 1] matter here?

    • They show integrals always swap with limits
    • They show pointwise limits can break continuity
    • They show measurable functions stay continuous

  6. Indicator of a measurable set is a measurable function. Is this correct?

    Circle one:   True   False

  7. How to build simple functions increasing to a nonnegative f?

    • Take constant 0 sequence
    • Truncate at n and round down to multiples of 1 over n on dyadic pieces
    • Take decreasing constants to infinity
    • Use only continuous functions

  8. The set where f exceeds 3 is not measurable. What follows about f?

    • Nothing, since single levels never matter
    • That f is not measurable
    • That f must be discontinuous

  9. A student claims the limit of measurable functions might fail the preimage test. What is the flaw?

    • The limit rewrites with countable operations, which measurable sets survive
    • Every limit of functions is automatically continuous
    • Preimages are only defined for simple functions

  10. Function is the identity on the interval from 0 to 4. The set where output exceeds 1 has what measure?

    Answer: ______________

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Answer key

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

Functions that respect measurable sets W1-mt_VwNRjXA7ro-s1

  1. 3 · Length 4 minus 1 equals 3, times 1 gives 3.
  2. Check preimage of every open ray is measurable · Check preimage of every open ray is measurable is the criterion, while continuity, boundedness, and finiteness are neither necessary nor sufficient.
  3. That preimages of open rays are measurable sets · Measurability is a statement about preimages, and rays are enough to check.
  4. True · Countable operations on preimages keep the limit inside the class.
  5. They show pointwise limits can break continuity · Continuous powers converge pointwise to a jump at 1, so continuity does not survive.
  6. True · Preimages are empty, whole space, the set, or its complement, all measurable.
  7. Truncate at n and round down to multiples of 1 over n on dyadic pieces · Truncate at n and round down to multiples of 1 over n on dyadic pieces gives an increasing simple approximation, while the others do not converge up to f.
  8. That f is not measurable · One bad ray preimage already breaks the definition.
  9. The limit rewrites with countable operations, which measurable sets survive · Upper and lower limits turn the problem into countable unions and intersections.
  10. 3 · Outputs above 1 correspond to (1, 4], length 3.
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