properties for measure
Monotonicity: If , and , then .
If in , , and , then
For any in , we have
Subadditivity: If is a collection of sets from , then
Continuity from below: If is a collection of sets from such that for all , then
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
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).
- 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|
|Date of creation||2013-03-22 13:45:28|
|Last modified on||2013-03-22 13:45:28|
|Last modified by||matte (1858)|