The most important generalization of the concept of an
integral.
Let
be a space with a non-negative complete countably-additive measure
(cf.
Countably-additive set function;
Measure space),
where
.
A
simple function
is a
measurable function
that takes at most a countable set of values:
,
for
,
if
,
.
A simple function
is said to be
summable
if the series
converges absolutely (cf.
Absolutely convergent series);
the sum of this series is the
Lebesgue integral
A function
is summable on
,
,
if there is a sequence of simple
summable functions
uniformly convergent (cf.
Uniform convergence)
to
on a set of full measure, and if the limit
is finite. The number
is the
Lebesgue integral
This is well-defined: the limit
exists and does not depend on the choice of the sequence
.
If
,
then
is a measurable almost-everywhere finite function on
.
The Lebesgue integral is a linear non-negative functional on
with the following properties:
1)
if
and if
then
and
2)
if
,
then
and
3)
if
,
and
is measurable, then
and
4)
if
and
is measurable, then
and
In the case when
and
,
,
the Lebesgue integral is defined as
under the condition that this limit exists and is finite for any sequence
such that
,
,
.
In this case the properties 1), 2), 3) are preserved, but condition 4) is violated.
For the transition to the limit under the Lebesgue integral sign see
Lebesgue theorem.
If
is a
measurable set
in
,
then the Lebesgue integral
is defined either as above, by replacing
by
,
or as
where
is the characteristic function of
;
these definitions are equivalent. If
,
then
for any measurable
.
If
if
is measurable for every
,
if
and if
,
then
Conversely, if under these conditions on
one has
for every
and if
then
and the previous equality is true
(
-additivity
of the Lebesgue integral).
The function of sets
given by
is absolutely continuous with respect to
(cf.
Absolute continuity);
if
,
then
is a non-negative measure that is absolutely continuous with respect to
.
The converse assertion is the
Radon–Nikodým theorem.
For functions
the name
"Lebesgue integral"
is applied to the corresponding functional if the measure
is the
Lebesgue measure;
here, the set of summable functions is denoted simply by
,
and the integral by
For other measures this functional is called a
Lebesgue–Stieltjes integral.
If
,
and if
is a non-decreasing absolutely continuous function, then
If
,
and if
is monotone on
,
then
and there is a point
such that
(the
second mean-value theorem).
In
1902
H. Lebesgue
gave (see
[1])
a definition of the integral for
and measure
equal to the Lebesgue measure. He constructed simple
functions that uniformly approximate almost-everywhere on a set
of finite measure a measurable non-negative function
,
and proved the existence of a common limit (finite
or infinite) of the integrals of these simple functions as they tend to
.
The Lebesgue integral is a basis for various generalizations
of the concept of an integral. As
N.N. Luzin
remarked
[2],
property 2), called absolute integrability, distinguishes the Lebesgue integral for
from all possible generalized integrals.