proof of functional monotone class theorem

We start by proving the following version of the monotone class theorem.

Theorem 1.

Let (X,A) be a measurable spaceMathworldPlanetmathPlanetmath and S be a π-system ( generating the σ-algebra ( A. Suppose that H be a vector spaceMathworldPlanetmath of real-valued functions on X containing the constant functions and satisfying the following,

Then, H contains every bounded and measurable functionMathworldPlanetmath from X to R.

Let 𝒟 consist of the collectionMathworldPlanetmath of subsets B of X such that the characteristic function 1B is in . Then, by the conditions of the theorem, the constant function 1X is in V so that X𝒟, and 𝒮𝒟. For any AB in 𝒟 then 1BA=1B-1A, as is closed under linear combinationsMathworldPlanetmath, and therefore BA is in 𝒟. If An𝒟 is an increasing sequence, then 1An increases pointwise to 1nAn, which is therefore in , and nAn𝒟. It follows that 𝒟 is a Dynkin system, and Dynkin’s lemma shows that it contains the σ-algebra 𝒜.

We have shown that 1A for every A𝒜. Now consider any bounded and measurable function f:X taking values in a finite setMathworldPlanetmath S. Then,


is in .

We denote the floor function by . That is, a is defined to be the largest integer less than or equal to the real number a. Then, for any nonnegative bounded and measurable f:X, the sequence of functions fn(x)=2-n2nf(x) each take values in a finite set, so are in , and increase pointwise to f. So, f.

Finally, as every measurable and bounded function f:X can be written as the differencePlanetmathPlanetmath of its positive and negative parts f=f+-f-, then f.

We now extend this result to prove the following more general form of the theorem.

Theorem 2.

Let X be a set and K be a collection of bounded and real valued functions on X which is closed under multiplication, so that fgK for all f,gK. Let A be the σ-algebra on X generated by K.

Suppose that H is a vector space of bounded real valued functions on X containing K and the constant functions, and satisfying the following

  • if f:X is bounded and there is a sequence of nonnegative functions fn increasing pointwise to f, then f.

Then, H contains every bounded and real valued A-measurable function on X.

Let us start by showing that is closed under uniform convergence. That is, if fn is a sequence in and fn-fsupx|fn(x)-f(x)| convergesPlanetmathPlanetmath to zero, then f. By passing to a subsequence if necessary, we may assume that fn-fm2-n for all mn. Define gnfn-21-n+2+f. Then gn since is a vector space containing the constant functions. Also, gn are nonnegative functions increasing pointwise to f+2+f which must therefore be in , showing that f as required.

Now let 0 consist of linear combinations of constant functions and functions in 𝒦 and ¯0 be its closureMathworldPlanetmathPlanetmath ( under uniform convergence. Then ¯0 since we have just shown that is closed under uniform convergence. As 𝒦 is already closed under productsPlanetmathPlanetmath, 0 and ¯0 will also be closed under products, so are algebras ( In particular, p(f)¯0 for every f¯0 and polynomialPlanetmathPlanetmath p[X]. Then, for any continuous functionMathworldPlanetmathPlanetmath p:, the Weierstrass approximation theoremMathworldPlanetmath says that there is a sequence of polynomials pn converging uniformly to p on bounded intervals, so pn(f)p(f) uniformly. It follows that p(f)¯0. In particular, the minimum of any two functions f,g¯0, fg=f-|f-g| and the maximum fg=f+|g-f| will be in ¯0.

We let 𝒮 consist of the sets AX such that there is a sequence of nonnegative fn¯0 increasing pointwise to 1A. Once it is shown that this is a π-system generating the σ-algebra 𝒜, then the result will follow from theorem 1.

If fn,gn¯0 are nonnegative functions increasing pointwise to 1A,1B then fngn increases pointwise to 1AB, so AB𝒮 and 𝒮 is a π-system.

Finally, choose any f𝒦 and a. Then, fn=((n(f-a))0)1 is a sequence of functions in ¯0 increasing pointwise to 1f-1((a,)). So, f-1((a,))𝒮. As intervals of the form (a,) generate the Borel σ-algebra on , it follows that 𝒮 generates the σ-algebra 𝒜, as required.

Title proof of functional monotone class theorem
Canonical name ProofOfFunctionalMonotoneClassTheorem
Date of creation 2013-03-22 18:38:44
Last modified on 2013-03-22 18:38:44
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 28A20