2.6 NONMEASURABLE SETS
We have dened what it means for a set to be measurable
and studied properties of the collection of measurable sets. It
is only natural to ask if, in fact, there are any sets that fail to
be measurable. The answer is not at all obvio
4.6 UNIFORM INTEGRABILITY: THE VITALI
CONVERGENCE THEOREM
We conclude this rst chapter on Lebesgue integration by
establishing, for functions that are integrable over a set of
nite measure, a criterion for justifying passage of the limit
under the integra
4.5 COUNTABLE ADDITIVITY AND CONTINUITY OF
INTEGRATION
The linearity and monotonicity properties of the Lebesgue
integral, which we established in the preceding section, are
extensions of familiar properties of the Riemann integral. In
this brief section
4.4 THE GENERAL LEBESGUE INTEGRAL
For an extended real-valued function f on E , we have
dened the positive part f + and the negative part f of F ,
respectively, by
f + (x ) = max(f (x ), 0) and f (x ) = max(f (x ), 0) for all x E .
Then f + and f are nonn
4.3 THE LEBESGUE INTEGRAL OF A MEASURABLE
NONNEGATIVE FUNCTION
A measurable function f on E is said to be of nite support
if m(cfw_x E : f (x ) = 0) < . The set cfw_x E : f (x ) = 0 is
called the support of f .
In the preceding section, we dened the integ
4.2 THE LEBESGUE INTEGRAL OF A BOUNDED
MEASURABLE FUNCTION OVER A SET OF FINITE
MEASURE
The Dirichlet function, which was examined in the preceding
section, exhibits one of the principal shortcomings of the
Riemann integral: a uniformly bounded sequence o
4.1 THE RIEMANN INTEGRAL
We recall a few denitions pertaining to the Riemann
integral. Let f be a bounded real-valued function dened on
the closed, bounded interval [a, b ]. A partition
P = cfw_x0 , x1 , . . . , xn of [a, b ] is a nite subset of [a, b ]
3.3 LITTLEWOODS THREE PRINCIPLES, EGOROFFS
THEOREM, AND LUSINS THEOREM
J.E. Littlewoods three principles:
I. Every measurable set is nearly a nite union of intervals.
II. Every measurable function is nearly continuous.
III. Every pointwise convergent sequ
3.2 SEQUENTIAL POINTWISE LIMITS AND SIMPLE
APPROXIMATION
For a sequence cfw_fn of functions with common domain E and
a function f on E , there are several distinct ways in which it is
necessary to consider what it means to state that the
sequence cfw_fn
3.1 SUMS, PRODUCTS, AND COMPOSITIONS
All the functions considered in this chapter take values in the
extended real numbers. Recall that a property is said to hold
almost everywhere (abbreviated a.e.) on a measurable set E
provided it holds on E E0 , where
2.7 THE CANTOR SET AND THE CANTOR-LEBESGUE
FUNCTION
We have shown that a countable set has measure zero and a
Borel set is Lebesgue measurable. These two assertions
prompt the following two questions.
Question 1 If a set has measure zero, is it also count
5.1 UNIFORM INTEGRABILITY AND TIGHTNESS: A
GENERAL VITALI CONVERGENCE THEOREM
The Vitali Convergence Theorem requires that E have nite
measure.
Example For each natural number n, dene fn = [n,n+1]
and f 0 on R. Then cfw_fn is uniformly integrable over R