General course information
Homework assignments for Math 663
This will be updated as the semester progresses.
Note: No class on Thursday, October 4, due
to Convocation.
- Hand in exercises 4, 6, and 9 in Section 6.2.
- Hand in exercises 2, 3 in section 14.3.5.
- Also do the following problem: use the definition
of splitting and the fact that the collection of
splitting sets forms a sigma-field to show each
of the following:
- If A and B are splitting sets with |A| finite, then
|A-B| = |A| - |A intersect B|.
Is this statement true if |A| is infinite?
- If |A| and |B| are finite,
then |A union B| = |A|+|B| - |A intersect B|. Is this
statement true if either |A| or |B| is infinite?
- Hand in exercises 5, 7, 11, 14 in section 14.3.5.
- Also do (but don't hand in) exercises 8 and 16
in section 14.3.5.
- Hand in exercises 1, 3, 7, 10 in section 14.4.4.
- For extra credit, do the following problem: suppose fn
is a nonnegative sequence of functions on the interval [0,1]
such that Integral(fn(x) dx, x=0..1) converges to zero
as n converges to infinity. Show that there is a subsequence of
the fn which converges to zero pointwise almost
everywhere on the interval [0,1]. (Note that exercise 10 implies
that this result is false for the entire sequence, i.e. that it
is necessary to go to a subsequence). Hints: show that the measure
of the set where fn > epsilon converges to zero
as n -> infinity. Think about the relationship of this set
to the set where fn(x) does NOT converge to zero.