4
CHAPTER
I. The
integral
in an
abstract
space.
tegral
sign
were
not allowed
by
the
earlier
methods
of
integration.
The reason
for this is
to be
found
in the
complete additivity
of
Le-
besgue
measure.
The fundamental
theorems
ofLebesgue
(cf.
below
12)
stating
the
precise
circumstances
under
which
term
by
term
integration
is
permissible,
are
justly
regarded by
Ch.
J. de
la
Valise
Poussin
[I,
p.
44]
as
one
of
the finest results of
the
theory.
Lebesgue's
theory
of measure
has,
in
its
turn,
led
naturally
to
further
important generalizations.
Instead of
starting
with
area,
or
volume,
of
figures,
we
may imagine
a
mass
distributed
in
the
Euclidean
space
under
consideration,
and
associate
with
each
set
as its
measure,
(its
"weight"
according
to Ch.
J.
de
la Valise
Poussin
[I,
Chap.VI;
1]),
the amount of mass
distributed
on the
set.
This,
again,
leads
to
a
generalization
of
the
integral,
parallel
to
Lebesgue's,
known as the
Lebesgue-Stieltjes
integral.
In
order
to
present
a
unified
account
of
the
latter,
we
shall
consider
in
this
chapter
an
additive
class
of measurable sets
given
a
priori
in
an
arbitrary
abstract
space.
We
shall
suppose
further,
that
in
this
class,
a
completely
additive measure
is determined
for the
mea-
surable sets. These
hypotheses
determine
completely
a
corresponding
method
of
integration
in the
Lebesgue
sense.
All the essential
prop-
erties
of the
ordinary
Lebesgue
integral,
except
at
most
those
im-
plying
the
process
of
derivation,
remain
valid
for
this
abstract
integral.
From this
point
of
view,
in
a
more or
less
general form,
the
Lebesgue
integral
has been
studied
by
a
number
of
authors,
among
whom
we
may
mention J.
Radon
[1],
P.
J.
Daniell
[2],
O.
Nikodym
[2]
and
B.
Jessen
[1].
For
further
generalizations
(of
a
somewhat
different
kind)
see
also A.
Kolmogoroff [1],
S.
Bochner
[1],
G. Fichtenholz
and L.
Kantorovitch
[1],
and M. Gowu-
rin
[1].
$
2.
Terminology
and
notation. Given
two
sets
A and
B,
we
write A
B
when
the
set A
is
a
subset
of
the
set
,
i. e.
when
every
element
of
A
is
an
element
of B.
When
we
have
both A
C
B
and
BA,
i.
e.
when
the
sets A
and
B
consist
of the
same ele-
ments,
we
write
A
=
B.
Again,
a
eA
means
that
a
is
an element
of
the
set
A
(belongs
to
A).
By
the
empty set,
we mean
the
set
without
any element;
we
denote
it
by
0.
A
set A
is
enumerable
if
there
existft
an
infinite
sequence
of distinct
elements o
1?
a
2
, .,.,
a,,,
...
consisting
of
all the
elements
of
the
set
A.
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