If is a probability space and is a measurable space (i.e., a set along with a sigma-algebra on ), then a *random variable* is a measurable function . That is, for each , we have . Generally speaking, we’ll be taking to be where is the *Borel algebra* on .

Given a topological space , there exists a sigma-algebra , called the Borel algebra on , that contains all open sets (members of ) and is the smallest such sigma-algebra. This means that for each we have and also if is another sigma-algebra on with this property, then . In general, given a collection of subsets of , there exists a sigma-algebra, which we’ll call , that is the smallest sigma-algebra on (smallest using as the ordering relation) containing each ; we say that is the sigma-algebra *generated* by . In this sense, the Borel algebra on is the sigma-algebra generated by the topology . We will usually just write to mean .

In the special case of , there’s an easy way to check to see that a function is a random variable (or a measurable function in general): we just look at the pre-images of intervals. Since the pre-images of functions are well-behaved with respect to set operations like union, intersection, and complement, it in fact suffices to only consider pre-images of the form . That is, if we show that for every , we will have shown that is measurable, and so a random variable. (Actually, we can look at all intervals of the form , , or . Using properties of sigma-algebras we can easily show that if we have all intervals of any of these forms, we have all intervals of any other form. Again, using properties of sigma-algebras it’s easy to take that and show that we have all countable unions of intervals — namely all countable unions of open intervals, i.e., all open sets.)

To see this, suppose for every we have .

So now we have the pre-images of all intervals of the form and . If we can also get pre-images for the form , it’ll be an easy jump to countable unions of open intervals. Note that

Now we easily see that

And we have pre-images of intervals of the form . Combined with the fact we have intervals of the form , it’s easy to see that we also have intervals of the form . Using properties of sigma-algebras, it’s easy to show now that we have the pre-image of any open set. This tells us that if for every , then we have that is measurable, and so a random variable.

In the case that is a countable set, we take to be (the powerset of ), and so *any* function is a random variable. This is because the pre-image of must be something (even if it’s empty); we have for every that , and so every function on a countable sample-space is a random variable.

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