lU. J. Mh.
1978) 69-14
Math. S.
69
A NOTE ON RIEMANN INTEGRABILITY
G. A. BEER
Department of Mathematics
California State University
Los Angeles, California
90032
(Received November 21, 1977)
ABSTRACT.
In this note we define Riemann integrabillty for real valued
functions defined on a compact metric space accompanied by a finite Borel
measure.
If the measure of each open ball equals the measure of its corre-
sponding closed ball, then a bounded function is Riemann integrable if and
only if its set of points of discontinuity has measure zero.
Let
denote the algebra of sets generated by the open and closed
subintervals of an interval
defined on
[a,b]
[a,b].
A bounded real valued function
is Riemann integrable if for each positive
exist two functions
and
istic functions of sets in
f
, there
that are linear combinations of character-
{
fba , dm- sba
satisfying
<. f
<_
and
dm <
where m denotes ordinary Lebesgue measure.
Riemann integrability may be
defined in an analagous way for real valued functions defined on a compact
metric space
K
accompanied by a finite Borel measure.
If we make a simple
G.A. BEER
70
assumption about the balls of
K, then the following famous theorem of
a bounded real valued function
Lebesgue extends:
f
[a,b]
defined on
f
Riemann integrable if and only if the set of points at which
is
is not
continuous has Lebesgue measure zero.
Suppose that
Let
K.
measure on
K
is a compact metric space and
{y- d(x,y)
Br(X
< r}
denote the open and closed balls of radius
and
is a finite Bore1
B--r(X )
in
K.
Any element of
denote the algebra generated by all such balls.
Let
x
about a point
r
r}
{y" d(x,y)
is of the form
1
where
19
i
.
m
0
Aik
k n.
(i)
{m,n l,...,nm
determined by elements of
where each
1
di
x.
has the form
Hence a step function
is real and
Sinceis
A. 6
1
are step functions, then so are
an algebra
It is easy to see that if
can be taken to be pairwise disjoint.
and
are positive
A step function is a linear combination of characteristic functions
integers.
{A i}
S
is a ball or its complement and
Aik
diXA.
I
+
, ,
the
#
inf {,}, and
sup {,}.
DEFINITION.
A bounded real valued function
_< f <_
and
d
d
Given a bounded real valued function
envelope
h
of
h(x)
f
K
is Riemann
and
there exist step functions
integrable if for each positive
such that
defined on
f
< e.
f
defined on
K, the upper
is the function defined by
infs>oSUpy 6 Bs{x)
f(Y)
x
K
$
RI EMANN INTEGRABILITY
and the lower envelope
g(x)
g
of
f
h
g(x) <_ f(x) <_ h(x)
only
is continuous at
f
f(Y)
xC K
is upper semicontinuous,
continuous,
if
is defined by
sup>0infy C B(x)
It is well known that
71
THEOREM.
Suppose (B r(x))
positive
r.
for each
x
x, and
(x)
h(x)
if
and
(see Royden [1, p.49]).
(Br(x))
for each
A bounded real valued function
f
x
in
K and for each
K
defined on
integrable if and only if the set of points at which
has
is lower semi-
g
f
is Riemann
is discontinuous
v-measure zero.
Proof.
Let h be the upper envelope of f and
Let
be any step function that exceeds
f.
g
its lower envelope.
Since each member of
d
can
be expressed in the form depicted in (1), the condition on the balls of
implies that each member of
u-measure zero.
{
It follows that
n
E
where
(i)
m < j <_ n
is the union of an open set and a set of
can be represented as
a
XA"
is an open set for
A.3
(iii)
{AI,A2
n}
A
1 S j <_ m
(ii)
for
0
u(Aj)
K.
partition
m
Let
such that
x C [3 A..
j=l J
(x) >
v{x" (x) < h(x)}
Since
supy C B6(x)
is constant near
f(Y)
O, and we have
so that
f
dv
>
x, there exists
(x) >. h(x).
f h du.
Hence,
We now construct a
decreasing sequence of step functions converging pointwise to
h
so that
K
G.A. BEER
72
inf
{
Let
described by
0
N.
be
h(xi)
eN(X
+
inf
{h(xi)
Given any positive integer
{eN+p,N+p_I}.
N+p > N+p+l > h
inf
and
Now let
I/N.
8N’K
d(xo,Xn)
K and
in
so large that
1/n < e.
1/n and
<_
<
eN+p as
p N+p
for each
N+p(XO) > h(xo)
n
Br.(Xi)}.
1
Define
,n(Xo)
above
> 0
+
to be
and let
@N+p
is a step function,
suppose to
we have for each p
+ 2e
xn
There exists a point
h(Xn)
Br. (x i),
@N
To establish the pointwise convergence
x0
be
R be the step function
/
p, define
Clearly,
the contrary that for some
Pick
I/N such that if y 6
I/N" x C
+
{Brl(Xl)’’’" ’Brm (Xm) }
Let
K by balls of radius at most
h(y) <
I h d.
is a step function}
and
N be a fixed positive integer.
a cover of
then
>_ f
d"
1/n.
h(Xn)
Clearly,
,h. Hence,
which violates the upper semicontinuity of
{n }
such that
>
h(xo)
+ e
is the
desired sequence.
Using the above technique we can show in the same manner that
g d
sup
{
and
f
d"
is a step function}.
The proof is
now completed by observing the equivalence of the following statements;
(i)
f
is Riemann integrable
(ii)
g d
continuous except at a set of points of
Ihd
(iii)
f
is
-measure zero.
A simple example shows that the theorem need not hold if our condition
on the balls of the metric space is omitted.
disc in the plane with the usual metric.
define
(B)
to be
I(B
{(x,y)" x 2 +
If
y2
K be the closed unit
Let
B
i})
is a Borel subset of
+
2
K,
{Bf {(x,y)" x 2 + y 2
RIEMANN INTEGRABILITY
< 1}
where
2
is two dimensional Lebesgue measure and
73
B1
is one
dimensional Lebesgue measure, considering the circle as having measure
2.
Then the characteristic function of the unit circle is Riemann
integrable (being a step function), but its set of discontinuities has
measure 2.
REFERENCES
I.
H. L. Royden.
Real Analysis, Macmillan, New York, 1968.
KEY WORDS AND PHRASES. Riemann integrable functions on a compact mric
space, Compact metric space with Borel measure.
AMS(MOS) SUBJECT CLASSIFICATIONS (1970).
28A25.