Upper and Lower Ordinate Sets
Let be a measurable space so that itself is measurable — that is, so that is a -algebra — and let be the real line with the -algebra of Borel sets.
If is a real-valued, non-negative function on , then we define the “upper ordinate set” to be the subset such that
We also define the “lower ordinate set” to be the subset such that
We will explore some basic properties of these sets.
First, if is the characteristic function of a measurable subset , then is the measurable rectangle , while is the measurable rectangle .
Next, if is a non-negative simple function, then we can write it as the linear combination of characteristic functions of disjoint subsets. If , then for each the upper ordinate set is the measurable rectangle while the lower ordinate set is the measurable rectangle . Since the are all disjoint, the upper ordinate set is the disjoint union of all the , and similarly for the lower ordinate sets. Thus the upper and lower ordinate sets of a simple function are both measurable.
Next we have some monotonicity properties: if and are non-negative functions so that for all , then and . Indeed, if then , so as well, and similarly for the lower ordinate sets.
If is an increasing sequence of non-negative functions converging pointwise to a function , then is an increasing sequence of sets whose union is . That is increasing is clear from the above monotonicity property, and just as clearly they’re all contained in . On the other hand, if , then But since increases to , this means that for some , and so . Thus is contained in the union of the . Similarly, if is decreasing to , then decreases to .
Finally (for now), if is a non-negative measurable function, then and are both measurable. The lower ordinate set is easier, since we know that we can pick an increasing sequence of non-negative measurable simple functions converging pointwise to . Their lower ordinate sets are all measurable, and they form an increasing sequence of measurable sets whose union is . Since this is a countable union of measurable sets, it must be itself measurable.
For we have to be a little trickier. First, if is bounded above by , then is non-negative and also bounded above by , and we can find an increasing sequence of non-negative measurable simple functions converging pointwise to . Then is a decreasing sequence of non-negative simple functions converging pointwise to . The catch is that the measurability of a simple function only asks that all the nonzero sets on which it is defined be measurable. That is, in principle the zero set of may be non-measurable. However, the zero set of is the complement of , and since this set is measurable we can use the assumed measurability of to see that is measurable as well. And so we see that is measurable as well. Thus is a decreasing sequence of measurable sets, converging to , which must thus be measurable.
Now, for a general , we can consider the sequence which replaces any value with . Each of these functions is still measurable (again using the measurability of ), and is now bounded. Thus is an increasing sequence of measurable sets, and I say that now their union is . Indeed, each is contained in , so the union must be. On the other hand, if , then . But since , there is some so that . Thus , and so , and is in the union as well. Since is the union of a countable sequence of measurable sets, it is itself measurable.
Incidentally, this implies that if is a non-negative measurable function, then the difference is measurable. But we can calculate this difference as
That is, is exactly the graph of the function , and so we see that the graph of a non-negative measurable function is measurable.