1. ## Monotone class theorem

Hi !

Now it's the turn of measure theory

I don't understand something I've read...
We have a sequence of sigma-algebras $\mathcal{F}_n$.
And $\mathcal{F}$ is the smallest sigma-algebra containing every $\mathcal{F}_n$

There's a property one wants to prove for any $A_1,\dots,A_k$ in $\mathcal F$

And it is said that

Assume we can prove the desired identity for all n and for all $A_1,\dots,A_k\in\mathcal F_n$.
By the monotone class theorem, the identity holds true for any $A_1\in \mathcal F$ and all n and all $A_2,\dots,A_k\in\mathcal F_n$ and so on...
I'm not sure : can we write that $\mathcal F=\bigcup_{n\geq 1} \mathcal{F}_n$ ? I think an uncountable union of sigma-algebras is a sigma-algebra, but I still have some doubts. Otherwise, I know I can just say that $\mathcal{F}$ is generated by the union.

Next point, I don't understand how the monotone class theorem acts here... I followed a link in the Wikipedia to transfinite induction. I have an intuition that the explanation has something to do with that, but I'm sorry to say that I didn't understand much in it, or rather how to use it here...

So is anyone able to explain it to me, please ? ^^

Thank you

2. The first thing, if $\mathcal F$ is the SMALLEST $\sigma$-algebra containing all $\mathcal F_n$ , then $\mathcal F$ should be

$
\mathcal F=\bigcap_n \mathcal F_n$
.

Maybe they mean Dynkin's lemma? sometimes it is mistakingly called "the monote class theorem". Do you know how this routine application of the lemma works in measure and probility theory??

Let me check on the rest and ill get back to you.

3. Hi,

If I'm not incorrect, it's true that Dynkin's lemma and the monotone class theorem are similar in their phrasings. But the paper I'm reading is globally very well written, so I don't know. Maybe there's just some subtlety.

Anyway, whatever it is - Dynkin or monotone class, I've never used any of them in a problem, and the only times I saw them was for proving some theorems from which one didn't even have to learn the proof...

As for the smallest sigma-algebra, suppose $\mathcal F_n\cap \mathcal F_{n-1}=\emptyset$... I believe it should be a union, or the sigma-algebra generated by the union...

Thanks for your reply, anything that can make me think is welcomed

4. But let me get this straight, are we talking about a sequence of sigma algebras of a certain set, lets say, X ?

In that case then they cant be mutually excluyent since at least

$\mathcal F_{l}\cap \mathcal F_{r}=\{\emptyset,X\}$ for every l,r

thus the worst scenerio is when $\mathcal F=\{\emptyset,X\}$.

Check the section about "generated sigma algebras" from:
http://en.wikipedia.org/wiki/Sigma-a....CF.83-algebra

-------------------------------------------
Now let me tell you how dynkin's lemma is usually used and maybe we can work this out.

Suppose $(X,\mathcal F)$ is a measurable space and $\mathcal F=\sigma(\mathbb A)$ where $\mathbb A$ is a family of subsets of X.

Suppose every set of $\mathbb A$ satisfy property "p". The use of dynkin's lemma is prove that the whole $\mathcal F$ satisfy "p".
The next part is routine:

--Check that $\mathbb A$ is a $\pi$-system.
This means check that it is closed under intersection. (In many cases $\mathbb A$ turns out to be an algebra of sets, thus closed under inteserction.)

Define $L=\{B\in 2^X: B \text{ satisfies property "p"}\}$

--Check that L is a $\lambda$-system.
This means:
1) $X\in L$

2)If $C,D \in L , C\subset D$ then $D-C \in L$ (Closed under complement)

3) $C_1\subset C_2 \subset\ldots \in L$ then $\bigcup C_i \in L$ (Closed under monote increasing sequences)

Then by Dynkin's lemma the sigma algebra generated by the $\pi$-system is contained on the $\lambda$-system, in symbols:

$\mathcal F=\sigma(\mathbb A)\subset L$

thus every set of $\mathcal F$ satisfy "p".

------

As i said, this is a pretty standard way to proving properties for sigma algebras knowing they are satisfy in a smaller family of sets (generators). In fact, in probability ,where often many sigma algebras of the same space play an important paper, dynkin's lemma is of paramount usefulness.

Still thinking about how to use it.

5. Originally Posted by mabruka
But let me get this straight, are we talking about a sequence of sigma algebras of a certain set, lets say, X ?

In that case then they cant be mutually excluyent since at least

$\mathcal F_{l}\cap \mathcal F_{r}=\{\emptyset,X\}$ for every l,r

thus the worst scenerio is when $\mathcal F=\{\emptyset,X\}$.

Check the section about "generated sigma algebras" from:
Sigma-algebra - Wikipedia, the free encyclopedia
Yeah, that was stupid, I forgot that 0 (empty set) and X always belong to a sigma-algebra...
But sorry, I still don't agree with that... I'm well aware what a generated sigma-algebra is.
And if we're talking about generated s-a by a family of sets, the only place where an intersection appears is when we define this s-a to be the intersection of all the s-a that contain this family.
Here, we're talking about the smallest s-a that contains every $\mathcal F_n$.

For example, (0 stands for empty set) let X={1,2,3}, let F1={0,{1},{2,3},X} and F2={0,{2},{1,3},X}
Their intersection is indeed {0,X}
But the smallest sigma-algebra that contains F1 and F2 will contain 0,{1},{2},{1,3},{2,3},X, but also {1}U{2}={1,2} and its complement {3}
(it's the power set of X)

There was something strange between these two theorems... If I look for the "monotone class theorem" in the French Wikipedia, its corresponding article in the English Wikipedia will be Dynkin's theorem.
But there exists a seperate "monotone class theorem" in the English Wikipedia, which looks similar, but a bit different from Dynkin's and which is the one I had in my notes in measure theory...

The main problem is that the guy who wrote the paper is a Chinese who teaches in France, and speaks French, but who wrote in English
I don't know which one he refers to.

Plus, I still don't understand why you're convinced it's Dynkin's theorem... The monotone class theorem (assuming it's this one : http://en.wikipedia.org/wiki/Monotone_class_theorem - algebra = stable by finite union and intersection) states that there's equality of the sigma-algebra and the monotone class. Well, it looks stronger, for not so many differences in the statement oO

6. Hmm you are right about the intersection! It is the intersection of al lthe sigma algebras containing the family! My bad!

Also, i never said i was convined that dynkin's lemma was used there, i only said that since dynkin's lemma its used frequently and has relation (sometimes called) monone calss theorem then MAYBE it was used there, but how i wouldnt know yet. I gave the context on how it is used to help us decide wheter or not it was used, ow how dynikns lemma or any similar theorem could be used.

It looks we will have to take a closer look, for start, to the one you post it there.