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For a sequence of ascending sets A1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · · its limit limn→∞ An is
defined as ∪n An . For a sequence of descending sets B1 ⊃ B2 ⊃ · · · ⊃ Bn ⊃ · · · , its limit
limn→∞ Bn is defined as ∩n Bn . For a general sequence of sets An , its upper limit set is
defined as
lim sup An = lim (∪n≥k An ) ,
n
k→∞
and its lower limit set is defined as
lim inf An = lim (∩n≥k An ) .
n
k→∞
1. Let A1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · · be a sequence of ascending measurable sets, show
that
m( lim An ) = lim m(An ).
n→∞
n→∞
2. Let B1 ⊃ B2 ⊃ · · · ⊃ Bn ⊃ · · · be a sequence of descending measurable sets, suppose
that m(B1 ) < ∞, show that
m( lim Bn ) = lim m(Bn ).
n→∞
n→∞
Show by examples that the condition m(B1 ) < ∞ cannot be removed in general.
P
3. Let {An } be a sequence of measurable sets, suppose ∞
n=1 m (An ) < ∞, prove that
m (lim supn An ) = 0.
4. In Stein’s book, A is called measurable if ∀ > 0, there exists an open set O ⊃ E
such that m∗ (O\E) < . Prove this definition is equivalent to the one using Carathéodory
criterion.
5. Let A1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · · be a sequence of ascending sets, show that
m∗ ( lim An ) = lim m∗ (An ).
n→∞
n→∞
(Hint: for any set E, there exists a Gδ set G containing E such that m∗ (E) = m(G).)
6. If E and F are two measurable subsets of real numbers with positive measure, show
E + F contains an interval.
7. Given two measurable sets A and B in Rn , their Minkowski sum is
A + B = {a + b | a ∈ A, b ∈ B}.
Now suppose A + B is also measurable, use the following steps to prove the BrunnMinkowski inequality:
1
1
1
m(A + B) n ≥ m(A) n + m(B) n .
1
2
(1) Prove B-M inequality for two open rectangles.
(2) Prove B-M inequality for the case that A = A1 ∪ · · · ∪ Ai and B = B1 ∪ · · · ∪ Bj
are disjoint union of open rectangles.
(3) Prove B-M inequality for open sets.
(4) Prove B-M inequality for general measurable sets.