properties for measure
Theorem [1, 2, 3, 4]
Let be a measure space, i.e.,
let be a set, let be a -algebra of sets
in , and let be a measure
on .
Then the following properties hold:
-
1.
Monotonicity: If , and , then .
-
2.
If in , , and , then
-
3.
For any in , we have
-
4.
Subadditivity: If is a collection
of sets from , then
-
5.
Continuity from below: If is a collection of sets from such that for all , then
-
6.
Continuity from above: If is a collection of sets from such that , and for all , then
Remarks In (2), the assumption assures
that the right hand side is always well defined, i.e., not of
the form . Without the assumption we can prove that
(see below).
In (3), it is tempting to
move the term to the other side for aesthetic reasons.
However, this is only possible if the term is finite.
Proof. For (1), suppose . We can then
write as the disjoint union , whence
Since , the claim follows. Property (2) follows from the above equation; since , we can subtract this quantity from both sides. For property (3), we can write , whence
If is infinite, the last inequality
must
be equality, and either of or must be infinite.
Together with (1), we obtain that if any of the quantities
or is infinite,
both sides in the equation are infinite and the claim holds.
We can therefore
without loss of generality assume that all quantities are finite.
From , we have
and thus
For the last two terms we have
where, in the second equality we have used properties for the
symmetric set difference (http://planetmath.org/SymmetricDifference), and
the last equality follows from
property (2). This completes the proof of
property (3).
For property (4), let us define the sequence
as
Now for , so is a sequence of disjoint sets. Since , and since , we have
and property (4) follows.
TODO: proofs for (5)-(6).
References
- 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
- 2 A. Mukherjea, K. Pothoven, Real and Functional analysis, Plenum press, 1978.
- 3 D.L. Cohn, Measure Theory, Birkhäuser, 1980.
-
4
A. Friedman,
Foundations of Modern Analysis
, Dover publications, 1982.
Title | properties for measure |
---|---|
Canonical name | PropertiesForMeasure |
Date of creation | 2013-03-22 13:45:28 |
Last modified on | 2013-03-22 13:45:28 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 8 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 60A10 |
Classification | msc 28A10 |