proof of functional monotone class theorem
We start by proving the following version of the 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 .
Let consist of the collection of subsets of such that the characteristic function is in . Then, by the conditions of the theorem, the constant function is in so that , and . For any in then , as is closed under linear combinations, and therefore is in . If is an increasing sequence, then increases pointwise to , which is therefore in , and . It follows that is a Dynkin system, and Dynkin’s lemma shows that it contains the -algebra .
We have shown that for every . Now consider any bounded and measurable function taking values in a finite set . Then,
is in .
We denote the floor function by . That is, is defined to be the largest integer less than or equal to the real number . Then, for any nonnegative bounded and measurable , the sequence of functions each take values in a finite set, so are in , and increase pointwise to . So, .
Finally, as every measurable and bounded function can be written as the difference of its positive and negative parts , then .
We now extend this result to prove the following more general form of the theorem.
Suppose that is a vector space of bounded 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 .
Let us start by showing that is closed under uniform convergence. That is, if is a sequence in and converges to zero, then . By passing to a subsequence if necessary, we may assume that for all . Define . Then since is a vector space containing the constant functions. Also, are nonnegative functions increasing pointwise to which must therefore be in , showing that as required.
Now let consist of linear combinations of constant functions and functions in and be its closure (http://planetmath.org/Closure) under uniform convergence. Then since we have just shown that is closed under uniform convergence. As is already closed under products, and will also be closed under products, so are algebras (http://planetmath.org/Algebra). In particular, for every and polynomial . Then, for any continuous function , the Weierstrass approximation theorem says that there is a sequence of polynomials converging uniformly to on bounded intervals, so uniformly. It follows that . In particular, the minimum of any two functions , and the maximum will be in .
We let consist of the sets such that there is a sequence of nonnegative increasing pointwise to . Once it is shown that this is a -system generating the -algebra , then the result will follow from theorem 1.
If are nonnegative functions increasing pointwise to then increases pointwise to , so and is a -system.
Finally, choose any and . Then, is a sequence of functions in increasing pointwise to . So, . As intervals of the form generate the Borel -algebra on , it follows that generates the -algebra , as required.
|Title||proof of functional monotone class theorem|
|Date of creation||2013-03-22 18:38:44|
|Last modified on||2013-03-22 18:38:44|
|Last modified by||gel (22282)|