Infinite Products, Part 1
Because we know that product spaces are product objects in the category of measurable spaces — at least for totally measurable spaces — we know that the product functor is monoidal. That is, we can define -ary products unambiguously as iterated binary products. But things start to get more complicated as we pass to infinite products.
If is a countably infinite collection of sets, the product is the collection of all sequences with for all . If each is equipped with a -ring and a measure , it’s not immediately clear how we should equip the product space with a -ring and a measure. However, we can give meaning to these concepts if we specialize. First we shall insist that each be a -algebra, and second we shall insist that be totally finite. Not just totally finite, though; we will normalize each measure so that .
This normalization is, incidentally, always possible for a totally finite measure space. Indeed, if is a totally finite measure on a space , we can define
It is easily verified that is another totally finite measure, and .
Now we will define a rectangle as a product of the form
where for all , and where for all but finitely many . We define a measurable rectangle to be one for which each is measurable as a subset of . We then define a subset of the countably infinite product to be measurable if it’s in the -algebra generated by the measurable rectangles. This defines the product of a countably infinite number of measurable spaces.
Now if is any set of positive integers, we say that two sequences and agree on if for all . A set is called a -cylinder if any two points which agree on are either both in or both out of . That is, membership of a sequence in is determined only by the coordinates for .
We also define the sets
so that we always have Each is itself a countably infinite product space. For every set and each point , we define the subset as the section of determined by . It should be clear that every section of a measurable rectangle in is a measurable rectangle in
Now if , and if E\subseteq X$ is a (measurable) -cylinder, then , where is a (measurable) subset of . Indeed, let be an arbitrary point of , and let be the -section of determined by this point. The sets (by assumption) and (by construction) are both -cylinders, so if belongs to either of them, then so does the point .
It should now be clear that if such a point belongs to either or , then it belongs to the other as well. Again using the fact that both and are -cylinders, if belongs to either or then so does the point . We can conclude that and consist of the same points. The parenthetical assertion on measurability follows from the fact that every section of a measurable set is measurable.