Monotone class theorem

Solution 1:

1) Monotone class theorem and $\pi-\lambda$-system of Dynkin are complementary ways to prove that a certain set of subsets contains a $\sigma$- algebra.

One can show that $M(G)$ the smallest monotone class of an algebra $G$ is a $\lambda$- system. Similarly, one can show that $\lambda(P)$ the smallest $\lambda$- system of a $\pi$- system $G$ is a monotone class.

The point is to see which is the simpler criterion.

Q:It is easier to check that $G$ is an algebra or to check that it is a $\pi$- system?

A: It is easier to check that $G$ is a $\pi$ system (every algebra is a $\pi$-system the converse does not follow)

Q:It is easier to check that $M$ is a monotone class or to check that it is a $\lambda$- system?

A: It is easier to check that $M$ is a $\lambda$ system (every algebra is a $\lambda$-system the converse does not follow)

2) To see that $\sigma(M) = \mathcal{F}_s$ note that you can approximate $1_{A_{s_1}}(X_1) 1_{A_{s_2}}(X_2) \ldots 1_{A_{s_k}}(X_k)$ with continuous functions $f_1(X_{s_1})f_2(X_{s_2})\ldots f_k(X_{s_k})$ (see https://en.wikipedia.org/wiki/Urysohn%27s_lemma)

the reason you choose the family of product of continuous functions is that one often deals better with properties for continuous functions.(in fact one may only need a denumerable set of continuous functions , depending on the problem at hand this is very useful)

note that one can know a measure by it's values on measurable sets ($\{\mu(A), A \in \mathcal{F}\}$) But one can also know a measure by it's values on continuous functions $\{\mu(f) = \int f \, d\mu, f \in C(X)\}$ when $X$ is a locally compact Hausdorff space. (see https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem)