Definition. Let be a set equipped with a
-algebra
. A measure on
(or on
, or simply on
if
such that
1). .
2). (Countable additivity) if is a sequence of disjoint sets in
, then
.
2)’. if are disjoint sets in
, then
,
because one can take for
. A function
that satisfies 1) and 2)’ but not necessarily 2) is called a finitely additive measure.
If is a set and
is a
-algebra,
is called a measurable space and the sets in
are called measurable sets. If
is a measure on
, then
is called a measure space.
Let be a measure space. If
(which implies that
for all
since
,
is called finite. If
where
and
for all
, the set
is said to be
-finite for
. More generally, if
where
and
for all
, the set
is said to be
-finite for
. If for each
with
there exists
with
and
,
is called semifinite.
Remark. Every -finite measure is semifinite, but not conversely.
Examples. 1) Let be any nonempty set,
, and
any function from
to
. Then
determines a measure
on
by the formula
.
is semifinite iff
for every
, and
is
-finite iff
is semifinite and
is countable.
a. If for all
,
is called counting measure;
b. If for some ,
is defined by
and
for
,
is called the point mass or Dirac measure at
.
2) Let be an uncountable set, and let
be the
-algebra of countable or co-countable sets. The function
on
defined by
if
is countable and
if
is co-countable is easily seen to be a measure.
3) Let be an infinite set and
. Define
if
is finite,
if
is infinite. Then
is a finitely additive measure but not a measure.
Theorem. Let be a measure space.
a). (Monotonicity) If and
, then $\mu(E)\le\mu(F)$.
b). (Subadditivity) If , then
.
c). (Continuity from below) If and
, then
.
d). (Continuity from above) If ,
, and
, then
.
Proof. a) If , then
.
b) Let and
for
. Then the
are disjoint and
for all
. Therefore, by a)
.
c) Setting , we have
.
d) Let ; then
,
, and
. By c) then
.
Since , we may subtract it from both sides to yield the desired result.
Remark. The condition in d) could be replaced by
for some
, as the first
‘s can be discarded from the sequence without affecting the intersection. However, some finiteness assumption is necessary, as it can happen that
for all
but
.
Definition. If is a measure space, a set
such that
is called a null set. By subadditivity, any countable union of null sets is a null set, a fact which we shall use frequently. If a statement about points
is true except for
in some every
. (If more precision is needed, we shall speak of a
-null set, or
-almost everywhere).
If and
, then
by monotonicity provided that
, but in general it need not be true that
. A measure whose domain includes all subsets of null sets is called complete. Completeness can sometimes obviate annoying technical points, and it can always be achieved by enlarging the domain of
, as follows.
Theorem. Suppose that is a measure space. Let
and
and
for some
. Then
is a
-algebra, and there is a unique extension
of
to a complete measure on
.
Proof. Since and
are closed under countable unions, so is
. If
where
and
, we can assume that
(otherwise, replace
and
by
and
). Then
, so
. But
and
, so that
. Thus
is a
-algebra.
If as above, we set
. This is well defined, since if
where $F_j\subset N_j\in\mathcal N$, then $E_1\subset E_2\cup N_2$ and so
, and likewise
. It is easily verified that
is a complete measure on
, and that
is the only measure on
that extends
.
Remark. The measure is called the completion of
, and
is called the completion of
with respect to
.
“Whenever you meet a ghost, don’t run away, because the ghost will capture the substance of your fear and materialize itself out of your own substance… it will take over all your own vitality… So then, whenever confronted with a ghost, walk straight into it and it will disappear.” ~ Allan Watts
References:
[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 24-27.
[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.