Mathematics 105, Spring 2004 — Problem Set VII
Remark
on problem 3.1.11:
If (
x
n
) is a sequence of real numbers,
lim
n
→∞
x
n
is a
common alternative notation for lim sup
n
→∞
x
n
; likewise lim
is denotes the lim inf.
In this problem analogous notions of lim sup and lim inf are defined for sequences
of sets. Inequality (i) deals with a relation between the lim inf of the real numbers
μ
(Γ
n
) and the
μ
measure of the lim inf of the sets Γ
n
; thus the first symbol lim
refers
to the notion for sets, while the second instance of the same symbol refers instead to
the notion for real numbers.
A couple of other miscellaneous remarks on the text:
(i) The discussion in
§§
3.1 and 3.2 is a bit more abstract and general than is really
necessary for us. In class I’ll discuss the main elements, in a streamlined way. I’ll
be perfectly satisfied if you master the topics discussed in lecture.
One example
is the concept of a measurable mapping. We’ll simply discuss measurable
R
valued
functions, that is, measurable mappings from some measure space to the extended real
numbers
R
= [
∞
,
+
∞
], and I’ll give a more direct definition. For present purposes
we won’t need to know about tensor products of measure spaces or of measurable
mappings (this will come up later, when we discuss the connection between Lebesgue
measure/integration on
R
n
,
R
m
, and
R
m
+
n
.
(ii) Wherever Stroock mentions topological spaces, you should assume that he is
talking about
metric
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 Spring '04
 MichaelChrist
 Topology, Real Numbers, measure, Lebesgue measure, Lebesgue integration

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