Stochastic processes can be seen as random variables taking their values in a function space. It is therefore important to understand the naturallly associated -algebras.

Let , , be the set of functions . We denote by the -algebra generated by the so-called cylindrical sets

where and where are products of intervals: .

As a -algebra is also generated by the following families:

- where and where are Borel sets in .
- where and where is a Borel set in .

**Exercise**: * Show that the following sets are not in :*

The above exercise shows that the -algebra is not rich enough to include natural events; this is due to the fact that the space is by far too big.

In these lectures, we shall mainly be interested in processes with continuous paths. In that case, we use the space of continuous functions endowed with the -algebra that is generated by

where

and where are products of intervals . This -algebra enjoys nice properties. It is for instance generated by the open sets of the (metric) topology of uniform convergence on compact sets.

**Proposition**

*The -algebra is generated by the open sets of the topology of uniform convergence on compact sets.*

**Proof**:

We make the proof when the dimension and let the reader adapt it in higher dimension. Let us first recall that, on the topology of uniform convergence on compact sets is given by the distance

This distance endows with the structure of a complete, separable, metric space (that is of a Polish space). Let us denote by the -field generated by the open sets of this metric space.

First, it is clear that the cylinders

are open sets that generate . Thus, we have

On the other hand, since for every , , and ,

we deduce that

Since is generated by the above sets, this implies

and concludes the proof.

**Exercise.**

Show that the following sets are in :

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Thank you Professor Baudoin for your wonderful website. May I ask a question. When defining the cylindrical sets, why do you have to use multiple n? If I define the cylindrical sets the same way as yours but only for n=1, wouldn’t they still generate the same sigma algebra? Thanks.

That’s a very good remark. If we consider cylindrical sets with n=1, they would actually generate the very same sigma-algebra because the cylinder with n >1 is an intersection of a finite number of cylinders with n=1.

Professor Baudoin, thank you very much for your reply. I am thinking about the first two exercise problems. Generally it is easier to show something is an element of a sigma-algebra, but how can I argue something is not in a sigma-algebra? I have a feeling that an element of the sigma-algebra T only has something to say for values of the functions on countably many t’s in [0,1], while sup f(t) < 1 poses restrictions on uncountably many t's. But how do I make the argument rigorous? Could you give a short hint?

We can define the distance d the same way on both spaces C and A (but then C is a Polish spaces while A is not, because A is not separable). Denote the sigma-algebras generated by cylinder sets in the two spaces by B and T respectively, then B is precisely the Borel sigma-algebra in C but T is not the Borel sigma-algebra in A (therefore we say B has nice properties in C while T is too small in A). Am I correct?

There are two typos in words “naturallly” and “indude”. And in the proof you use the term sigma-field instead of sigma-algebra.