infinite product measure
each is a totally finite measure, that is, .
In fact, we can now turn each into a probability space by introducing for each a new measure:
With the assumption that each is a probability space, it can be shown that there is a unique measure defined on such that, for any expressible as a product of with for all except on a finite subset of :
Then becomes a measure space, and in particular, a probability space. is sometimes written .
If is infinite, one sees that the total finiteness of can not be dropped. For example, if is the set of positive integers, assume and . Then for
would not be well-defined (on the one hand, it is , but on the other it is ).
The above construction agrees with the result when is finite (see finite product measure (http://planetmath.org/ProductMeasure)).
|Title||infinite product measure|
|Date of creation||2013-03-22 16:23:14|
|Last modified on||2013-03-22 16:23:14|
|Last modified by||CWoo (3771)|
|Defines||totally finite measure|