functional monotone class theorem
Let be a measurable space and be a -system (http://planetmath.org/PiSystem) generating the -algebra (http://planetmath.org/SigmaAlgebra) . Suppose that be a vector space of real-valued functions on containing the constant functions and satisfying the following,
for every set the characteristic function is in .
Then, contains every bounded and measurable function from to .
Here, is a bounded and -measurable function. The space of functions for which this identity holds is easily shown to be linearly closed and, by the monotone convergence theorem, is closed under taking monotone limits of functions. Furthermore, the -system of sets of the form for , generates the -algebra and
So, and the monotone class theorem allows us to conclude that all real valued and bounded -measurable functions are in , and equation (1) is satisfied.
An alternative, more general form of the monotone class theorem is as follows. The version of the monotone class theorem given above follows from this by letting be the characteristic functions for .
Suppose that is a vector space of real valued functions on containing and the constant functions, and satisfying the following
if is bounded and there is a sequence of nonnegative functions increasing pointwise to , then .
Then, contains every bounded and real valued -measurable function on .
Saying that is the -algebra generated by means that it is the smallest -algebra containing the sets for and Borel subset of .
For example, letting be as in theorem 2 and be finite measures on such that and for all , then for all bounded and measurable real valued functions . Therefore, . In particular, a finite measure on is uniquely determined by its characteristic function for . Similarly, a finite measure on a bounded interval is uniquely determined by the integrals for .
|Title||functional monotone class theorem|
|Date of creation||2013-03-22 18:38:36|
|Last modified on||2013-03-22 18:38:36|
|Last modified by||gel (22282)|