One of the first goals in constructing the theory of random processes, the foundations of measure theory having been built, is to show that we can often construct random sequences in a useful manner. Otherwise, it would be a rather dry field!
[This post borrows many ideas from Dr Bailleul's Probability Notes.]
Caratheodory’s Extension Theorem, a key piece of abstract machinery in measure theory, allows us to extend a countably additive set function, , on a ring, , to a measure on , the -algebra generated by .
Often, it’s fairly easy to show that a given set is a ring, and that a given function is an additive set function, and the main challenge is to prove -additivity.
Definition Given measurable spaces , we say a sequence of measures on is projective if, for all , .
Projective sequences are models of discrete-time random processes, with memory.
So, given a projective sequence of probability measures, we’d like to construct a probability on the infinite product -algebra on , representing our random sequence. Luckily enough, when our spaces are Borel spaces (very common, as we’ll see), we can do just this. Hurrah!
Definition We say a measurable space is Borel if it is isomorphic to a Borel subset of , with the corresponding -algebra.
So I suppose I should add…
Definition Analogously to topological spaces, we say that two measure spaces are isomorphic if there is a bijective measurable function between them, with a measurable inverse.
So, the main event, a theorem on the existence of random sequences, by Daniell.
Theorem Let be a sequence of Borel spaces and let be a projective sequence of probability measures on . Then, there’s a unique probability measure on the product -algebra of , such that, for all , for all , .
Proof We are trying to construct a probability measure, so let’s use Caratheodory’s Extension Theorem (for some tasks this can be, and should be, avoided).
We can quickly check that is a ring (do it!).
We can quickly check that generates the product -algebra on .
Define on by . This is well-defined since is a projective sequence.
We can quickly check that is an additive set-function on .
So, to use Caratheordory, we need to show is -additive. This constitutes the bulk of the proof. is -additive iff for every decreasing sequence with , . So let’s prove the contrapositive.
Suppose is a decreasing sequence of elements of , such that . WLOG, by inserting extra terms if necessary, assume , for some sequence .
is a Borel space, so there’s an isomorphism between this and a subset of . Let be the image measure of by . Fix . By inner regularity of with the Borel -algebra, we can find a compact set , such that .
Let . Then , so if we choose , then . In particular, , so we can pick .
By compactness of , the sequence formed by the projection of onto has a convergent subsequence for each . So, by a diagonalization argument, we can pass to a sequence whose limit is in for each . Thus , so we are done.
This post has wound up longer than expected, so I’ll stop here.