Simple and Elementary Functions
We now introduce two classes of functions that are very easy to work with. As usual, we’re working in some measurable space .
First, we have the “simple functions”. Such a function is described by picking a finite number of pairwise disjoint measurable sets and a corresponding set of finite real numbers . We use these to define a function by declaring if , and if is in none of the . The very simplet example is the characteristic function of a measurable function . Any other simple function can be written as
Any simple function is measurable, for the preimage is the union of all the corresponding to those , and is thus measurable.
It’s straightforward to verify that the product and sum of any two simple functions is itself a simple function — given functions and , we have and . It’s even easier to see that any scalar multiple of a simple function is simple — . And thus the collection of simple functions forms a subalgebra of the algebra of measurable functions.
“Elementary functions” are similar to simple functions. We slightly relax the conditions by allowing a countably infinite number of measurable sets and corresponding values .
Now, why do we care about simple functions? As it happens, every measurable function can be approximated by simple functions! That is, given any measurable function we can find a sequence of simple functions converging pointwise to .
To see this, first break up into its positive and negative parts and . If we can approximate any nonnegative measurable function by a pointwise-increasing sequence of nonnegative simple functions, then we can approximate each of and , and the difference of these series approximates . So, without loss of generality, we will assume that is nonnegative.
Okay, so here’s how we’ll define the simple functions :
That is, to define we chop up the nonnegative real numbers into chunks of width , and within each of these slices we round values of down to the lower endpoint. If , we round all the way down to . There can only ever be values for , and each of these corresponds to a measurable set. The value corresponds to the set
while the value corresponds to the set . And thus is indeed a simple function.
So, does the sequence converge pointwise to ? Well, if , then for all . On the other hand, if then ; after this point, and are both within a slice of width , and so . And so given a large enough we can bring within any desired bound of . Thus the sequence increases pointwise to the function .
But that’s not all! If is bounded above by some integer , the sequence converges uniformly to . Indeed, once we get to , we cannot have for any . That is, for sufficiently large we always have . Given an we pick an so that both and , and this will guarantee for every . That is: the convergence is uniform.
This is also where elementary functions come in handy. If we’re allowed to use a countably infinite number of values, we can get uniform convergence without having to ask that be bounded. Indeed, instead of defining for , just chop up all positive values into slices of width . There are only a countably infinite number of such slices, and so the resulting function is elementary, if not quite simple.