Math 318 HW #5
Due 5:00 PM Thursday, March 3
Reading:
Wilcox & Myers §10–12.
Problems:
1. Show that the sets Ex defined as part of showing that m∗ is not countably additive (cf.
Example 7.7) form equivalence classes. Use this to conclude that for any x, y ∈ [0, 1], either
Ex = Ey or Ex ∩ Ey = ∅.
2. Exercise 9.8.
3. Exercise 9.27.
4. Let S be the set of all intersections of Q with arbitrary closed, open, and half-open subintervals
of [0, 1], including degenerate closed subintervals consisting of a single point. Define µ : S → R
by
µ(Acd ) = d − c,
where Acd is the intersection of Q with any of the intervals (c, d), [c, d], [c, d), or (c, d]. Show
that µ is finitely additive but not countably additive and, hence, not a measure.
5. (a) Let S be a collection of subsets of [0, 1] that is closed under countable unions. Suppose
µ : S → R is a set function satisfying the first three conditions for being a measure. In
addition, suppose µ is
S
P
i. finitely additive (meaning µ(A) = ni=1 µ(Ai ) for A = ni=1 Ai with Ai ∈ S and
Ai ∩ Aj = ∅ whenever i 6= j); and
S∞
P
ii. countably sub-additive (meaning µ(A) ≤ ∞
i=1 Ai with Ai ∈ S
i=1 µ(Ai ) for A =
and Ai ∩ Aj = ∅ whenever i 6= j).
Prove that µ is a measure on S.
(b) Explain why Exercise 9.29 is an easy corollary of part (a).
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