A Primer of Lebesgue Integration

159 Pages·2001·5.19 MB·English

by  H. Bear

PREFACE TO THE FIRST EDITION
This text provides an introduction to the Lebesgue integral
for advanced undergraduates or beginning graduate students in
mathematics. It is also designed to furnish a concise review of
the fundamentals for more advanced students who may have
forgotten one or two details from their real analysis course and
find that more scholarly treatises tell them more than they want
to know.
The Lebesgue integral has been around for almost a century,
and the presentation of the subject has been slicked up con
siderably over the years. Most authors prefer to blast through
the preliminaries and get quickly to the more interesting results.
This very efficient approach puts a great burden on the reader;
all the words are there, but none of the music. In this text we
deliberately unslick the presentation and grub around in the
fundamentals long enough for the reader to develop some in
tuition about the subject. For example, the Caratheodory def
inition of measurability is slick—even brilliant—but it is not
intuitive. In contrast, we stress the importance of additivity for
the measure function and so define a set £ e (0,1) to be mea
surable if it satisfies the absolutely minimal additivity condi
tion: m(E) + m{E) = 1, where E^ = (0,1) — E and m is
the outer measure in (0,1). We then show in easy steps that
measurability of E is equivalent to the Caratheodory criterion,
m(E n T) + m(E^ oT) = m(T) for all T. In this way we remove
the magic from the Caratheodory condition, but retain its util
ity. After the measure function is defined in (0,1), it is extended
to each interval (n, n + 1) in the obvious way and then to the
whole line by countable additivity.
viii A PRIMER OF LEBESGUE INTEGRATION
We define the integral via the famihar upper and lower
Darboux sums of the calculus. The only new wrinkle is that
now a measurable set is partitioned into a finite number of
measurable sets rather than partitioning an interval into a fi
nite number of subintervals. The use of upper and lower sums
to define the integral is not conceptually different from the usual
process of approximating a function by simple functions. How
ever, the customary approach to the integral tends to create the
impression that the Lebesgue integral differs from the Riemann
integral primarily in the fact that the range of the function is par
titioned rather than the domain. What is true is that a partition
of the range induces an efficient partition of the domain. The
real difference between the Riemann and Lebesgue integrals is
that the Lebesgue integral uses a more sophisticated concept of
length on the line.
We take pains to show that both the Riemann-Darboux in
tegral and the Lebesgue integral are limits of Riemann sums,
for that is the way scientists and engineers tend to think of the
integral. This requires that we introduce the concept of a con
vergent net. Net convergence also allows us to make sense out
of unordered sums and is in any case something every young
mathematician should know.
After measure and integration have been developed on the
line, we define plane outer measure in terms of coverings by
rectangles. This early treatment of plane measure serves three
purposes. First, it provides a second example of the definition
of outer measure, and then measure, starting with a natural
geometric concept—here the area of a rectangle. Second, we
show that the linear integral really is the area under the curve.
Third, plane measure provides the natural concrete example of a
product measure and is the prototype for the later development
of general product measures.
The text is generously interlarded with problems. The prob
lems are not intended as an intelligence test, but are calculated
to be part of the exposition and to lure the reader away from a
passive role. In many cases, the problems provide an essential
step in the development. The step may be routine, but the reader
is nevertheless encouraged thereby to pause and become actively
PREFACE TO THE FIRST EDITION Ix
involved in the process. There are also additional exercises at
the end of each chapter, and the author earnestly hopes that
these will add to the reader's education and enjoyment.
The author is pleased to acknowledge the help of Dick
Bourgin, Bob Burckel, and Ken Ross, all of whom read the
manuscript with great care and suggested many improvements
in style and content.
PREFACE TO THE SECOND EDITION
The principal change from the first edition is the new one-shot
definition of the Lebesgue integral. The integral is first defined
for bounded functions on sets of finite measure, using upper
and lower Darboux sums for finite partitions into measurable
sets. This approach is designed to emphasize the similarity of
the Lebesgue and Riemann integrals. By introducing countable
partitions, we then extend the definition to arbitrary functions
(bounded or not) and arbitrary sets (finite measure or not). This
elegant touch, like many of my best ideas, was explained to me
by A. M. Gleason.
Many of the errors and crudities of the first edition have been
corrected, and the author is indebted to Robert Burckel, R. K.
Getoor, K. R S. Bhaskara Rao, Joel Shapiro, and Nicholas Young
for pointing out assorted mistakes. In addition, several anony
mous reviewers of the second edition made many helpful sug
gestions. I feel confident, however, that there remain enough
errors to challenge and reward the conscientious reader.
Finally, the author wishes to express his gratitude to Susan
Hasegawa and Pat Goldstein for their superb work with the
typing and proofreading.
H. S. Bear
May 2001
1
THE RIEMANN-DARBOUX INTEGRAL
We start by recalling the definition of the familiar Riemann-
Darboux integral of the calculus, which for brevity we will call
the Riemann integral. Our later development of the Lebesgue
integral will closely parallel this treatment of the Riemann inte
gral. We consider a fixed bounded interval [a, b] and consider
only real functions f which are bounded on [a,b].
A partition P of [^, ^] is a set P = {XQ, xi, Xi,..., Xn} of points
of [a, b] with
a = XQ < Xi < X2 < ' " < Xn = b.
Let /" be a given function on [a, b] with
w < f(x) < M
for all X e [a, b]. For each / = 1, 2,..., w let
mi = inf{ f(x) : X/_i < x < Xi},
Mi = sup{ f(x) : Xi-i < X < Xi).
In the usual calculus text treatment the infs and sups are taken
over the closed intervals [x/_i,X/]. In our treatment of the
Lebesgue integral we will partition [a, b] into disjoint sets, so we
use the disjoint sets (x/_i, Xi) here. We are effectively ignoring a
finite set of function values, f(xo), f(xi), fixi),..., f{Xn), and
in so doing we anticipate the important result of Lebesgue that
2 A PRIMER OF LEBESGUE INTEGRATION
function values on a set of measure zero (here the set of parti
tion points) are not relevant for either the Riemann or Lebesgue
integral.
The lewder sum L{f, P) and the upper sum U(f, P) for the
function f and the partition P are defined as follows:
n
L{f, P) = ^mi{Xi -x/_i),
i=l
Clearly m<mi< M/ < M for each /, so
m{b-a) < Uf,P) < U{f,P) < M(b-a).
For a positive function f on [a, b] the lower sum represents
the sum of the areas of disjoint rectangular regions which lie
within the region
S = {(x, y) :a <x<b,0 <y < f{x)].
Similarly, the upper sum U{f, P) is the area of a finite number
of disjoint rectangular regions which cover the region 5 except
for a finite number of line segments on the lines x = Xj. The
function f is said to be integrable whenever
sup L(/-,P) = infU(/-,?). (1)
p P
The integral of f over [a, b] is the common value in (1) and
is denoted /^ f. The area of S is defined to be this integral
whenever f is integrable and non-negative.
The integral is also sometimes written /^ f(x) dx^ particularly
when a change of variable is involved. The "x" in this expression
is a dummy variable and can be replaced by any variable except
f or d. For example,
/ f{x)dx = [ f(t)dt = f f(a)da = f f{c)dc.
J a J a J a J a
The last two versions are logically correct, but immoral, since
they flout the traditional roles oi a,b,c as constants, and the
1 THE RIEAAANN-DARBOUX INTEGRAL 3
third integral discourteously uses the same letter for the limit
and the dummy variable.
Now we make a few computations to derive the basic prop
erties of the integral and show that the integral exists at least
when f is continuous.
A partition Qis a refinement of the partition P provided each
point of P is a point of Q. We will indicate this by writing P <Q
without reference to the numbering of the points in P or Q.
Clearly Q is a refinement of P provided each of the subintervals
of [a, b] determined by Qis contained in one of the subintervals
determined by P.
Proposition 1. IfP<Q, then L( f, P) < L(/*, Q) andU( f, Q)<
U(f,P).
Proof. Suppose Q contains just one more point than P, and
to be specific assume this additional point x* lies between the
points xo and xi of P. If
m[ = inf{ f(x) : Xo < X < X*}
n/l = inf{ f(x) : X* < X < Xi}
mi = inf{ f(x) : Xo < X < Xi},
then 7f/^ > mi and m![ > mi so the sum of the first two terms in
L( f, Q) exceeds the first term of L( f, P):
m[(x^ — Xo) + m^[(xi — x*) > mi(xi — Xo).
The remaining terms of L( /*, Q) and L( /*, P) are the same, so
L(f, Q) > L(f, P). We can consider any refinement Q of P
as obtained by adding one point at a time, with the lower sum
increasing each time we add a point. The argument for the upper
sums is similar, ill
Proposition 2. Every lower sum is less than or equal to every
upper sum, as the geometry demands.
Proof. If P and Q are any partitions, and R = P U Q is the
common refinement, then
L{f, P) < L(f, R) < U(f, R) < U{f, Q). ill
4 A PRIMER OF LEBESGUE INTEGRATION
Proposition 3. f is integrable on [a, b] if and only if for each
8>0 there is a partition P of [a,b] such that U{fP) —
L{fP)<8.
Proof This useful condition is equivalent to sup L{f P) =
inf U( /*, P) in view of Proposition 2. ill
Proposition 4. If f is integrable on [a, b] and [a, ^] <z[a, b\
then f is integrable on[a, ^].
Proof Let s > 0 and let P be a partition of [a, b] such that
U(f P) — L(f P) < 6. We can assume that a and ^ are points of
P, since adding points increases L( /*, P) and decreases U(f P)
and makes their difference smaller. If PQ consists of the points
of P which are in [a, ^], then PQ is a partition of [a, jS]. Note
that
U{f P) - Uf P) = X](M, - m,)(x, ~ x,_i) (2)
and U( f PQ) — L( f PQ) is the sum of only those terms such that
Xi-i and Xi e PQ. Since we omit some non-negative terms from
(2)togetU(/',Po)-L(/*,Po),
U(f Po) - L{f Po) < U(f P) - Uf P) < 8, ill
Problem 1. If f is integrable on [a, b]^ then —f and | f\ are
integrable on [a, b], and Xf (-/") = -/.V, |xf/-| < Xf | f\. II
Problem 2. li a < c < b then / is integrable on [a, c] and on
[c, /?] if and only if f is integrable on [a,b]. In this case
rc pb rb
f+ f= f. •"•Hil l
Ja Jc Ja
Problem 3. If /"is integrable on [a, b] and g = /^ except at a fi
nite number of points, then g is integrable and f^ g = J^ f. ""HI
Problem 4. li a = XQ < xi < - -- < Xn = b and f is defined on
[a, b] with /"(x) = yi for x € (x/_i, X/), then f is integrable and
- x,_i). (Note that it is immaterial how f is
defined on the %/, by the preceding problem.) ""HI
1 THE RIEAAANN-DARBOUX INTEGRAL 5
Problem S. We say that g is a step function on [a, b] if there
is a partition a = XQ < Xi < xi < - - < Xn = b such that g
is constant on each (x/_i, X/). (By the preceding problem, step
functions are integrable with the obvious value for the integral.)
Show that if f is integrable on [a, b] there are step functions gn
and hn with {gn} increasing and {hn} decreasing and gn < f < hn
for all n and all x, and \im J^gn = HmJ^hn = Xf f- (Note: We
will be able to show later that gn alid hn approach f except
possibly on a set of measure zero.) ""HI
Proposition 5. If f is continuous on [a, b], then f is integrable
on [a, b\.
Proof. If f is continuous on [a,b\ then f is uniformly con
tinuous. Hence iis > 0 there is 5 > 0 so that \f{x) — f{x')\ < s
whenever \x — x^\ < S.li P is a partition with xi — Xi-\ < 8 for
all /, then M/ — mi < s for each /, so
U(f P) - L(f P) = J2(M, - m,){x, - x,_i)
< 8 Y^{Xi - Xi-i)
= s(b-a), ill
Problem 6. (i) If f is continuous on [a, b] except at a (or b)
and f is bounded on [a,b]^ then f is integrable on [a,b].
(ii) If f is bounded on [a, b] and continuous except at a finite
number of points, then f is integrable on [a,b].
(iii) Suppose f is bounded on [a, b] and discontinuous on a
(possibly infinite) set £. Assume that for each ^ > 0 there are
disjoint intervals (ai, b\),..., {a^, b^) contained in [a, b] such
that E c (ai, bi) U ' " U (a^, ^N) and TJiLxih ^ ^i) < ^- Show
that f is integrable. ""HI
So far we have the integral /^ f defined only when [a, b] is a
bounded interval and f is bounded on [a,b]. Now we extend
the definition to certain improper cases; i.e., situations where
the interval is unbounded, or f is unbounded on a bounded
interval. Typical examples of such improper integrals are
/ —z^dx and / -dx.
Jo ^x Jo 1 + x^
A PRIMER OF LEBESGUE INTEGRATION
In both these examples the integrand is positive and the defini
tion of the integral should give a reasonable value for the area
under the curve. The definitions of the integrals above are
/ —zz: dx = lim / —7= dx,
Jo y/x £-^0+ Js ^X
coo \ fb 1
/ :; :^ dx = lim / dx.
Jo 1 + X^ b-^oQ Jo 1 -\- X^
Both these limits are finite, so both functions are said to be
(improperly) Riemann integrable on the given interval. The
Lebesgue definition of the integral will give the same values.
In general, if f is integrable on[a + s.b] for all £ > 0, but f
is not bounded on [a, b\ (i.e., not bounded near a)^ we define
b
f= limj f,
J a s—^0+ Ja-\-s
provided this limit exists. Similarly, if f is integrable on every
interval [a, b] ior b > a we define
POO rb
J a b—^00 J a
when the Hmit exists. Similar definitions are made for Xf f if f
is unbounded near /?, and for /^^ /" if /* is integrable on [a, b]
for all a < b.
These definitions lend themselves to the calculations of ele
mentary calculus, but do not coincide with the Lebesgue defi
nition if f is not always positive or always negative. For ex
ample, if f is {—ly/n on [n,n + 1), n = 1,2,..., then f
is improperly Riemann integrable on [1, oc). We will see later
that f is Lebesgue integrable if and only if | /*| is Lebesgue in
tegrable. Hence the above function is not Lebesgue integrable
since ^ - = 00.
Problem 7. Show that /o~ '-^dx exists. "Hlll
Problem 8. (i) Exhibit a g on [0, oo) such that \g(x)\ = 1 and
/o~ g exists.