Monotonicity of Measures
First, we make a couple of definitions. A set function on a collection of sets
is “monotone” if whenever we have
and
in
with
, then
. We say that
is “subtractive” if whenever further
and
, then
.
Now I say that any measure on an algebra is both monotone and subtractive. Indeed, since
is an algebra, then
is guaranteed to be in
as well, and it’s disjoint from
. And so we see that
Since is non-negative, we must have
, and so
is monotone. And if
is finite, then we can subtract it from both sides of this equation to show that
is subtractive as well.
Next, say is a measure on an algebra
. If we have a set
and a finite or (countably) infinite sequence of sets
so that
, then we have the inequality
. To show this, we’ll invoke a very useful trick: if
is a sequence of sets in an algebra
, then there is a pairwise disjoint sequence
of sets so that each
and
. Indeed, we define
That is, is the same as
, and after that point each
is everything in
that hasn’t been covered already.
So we start with , and come up with a new sequence
. The (disjoint) union of the
is,
like the union of the
. Then, by the countable additivity of
, we have
. But the monotonicity says that
, and also that
, and so
.
On the other hand, if is a pairwise disjoint and we have
, then
. Indeed, if
is a finite sequence, then the union
is in
. Monotonicity and finite additivity then tell us that
However if is an infinite sequence, then what we just said applies to any finite subsequence. Then the sum
is the limit of the sums
. Each of these is less than or equal to
, and so their limit must be as well.