MAT 473
27
11.
Lebesgue Measure: Closed Boxes and Special Polygons
Recall that relatively few functions
f
:
R
→
R
are Riemann integrable: the continuous
functions, and in a sense not too many more.
Also, the Riemann integral behaves badly
with respect to limit operations — even pointwise limits of continuous functions need not
be Riemann integrable.
The Lebesgue integral improves on this. Very roughly speaking, the di
ff
erence between the
two is that the Riemann integral chops up the domain of the function, while the Lebesgue
integral chops up the range. A Riemann sum for
f
looks at a subinterval of the domain of
f
and asks how big
f
(
x
) is on that subinterval; a “Lebesgue sum” for
f
looks at a value
y
in the range of
f
and asks how big the set is on which
f
(
x
) is near
y
. This means we first
need to know how to suitably “measure” sets in
R
n
.
Definition 11.1.
Let
X
be a set. A
σ
algebra
of subsets of
X
is a collection
M
of subsets
of
X
such that
(i)
∅ ∈
M
.
(ii)
∪
∞
i
=1
A
i
∈
M
for any countable collection
{
A
i
}
⊆
M
.
(iii)
A
c
∈
M
for any
A
∈
M
.
It is immediate from the definition that if
M
is a
σ
algebra of subsets of
X
, then
X
∈
M
,
A
\
B
∈
M
for any
A, B
∈
M
, and
∩
∞
i
=1
A
i
∈
M
for any countable collection
{
A
i
}
⊆
M
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Polygons, DI, Riemann

Click to edit the document details