Simple functions are (almost) the most basic notion of measurable function in measure theory. Given a measure, it's easy to define the integral of a simple function, and we extend this to more general functions by continuity.
Let be a measurable space. We may want to be equipped with some more data; if is a measure space, then this is plenty of data. However, for the most basic definitions, it's enough if is simply a measurable space. This is the domain of our simple functions.
Another necessary datum is the simple functions' codomain , which we will eventually want to be at least a Banach space over the real numbers. (In the simplest example, is itself, or perhaps the space of complex numbers.) We take to be a measurable space using its Borel sets.
Since a simple function is measurable and a singleton is Borel, each fibre of is a measurable set in ; the function is given by the (finitely many) nonempty fibres and their (singleton) images. This suggests another way to look at simple functions:
Here we identify a measurable set with its characteristic function , so the formal linear combination is identified with the function , which is measurable and whose range is contained in the finite set of sums of the . (If there are terms in the linear combination, then there are at most such sums.)
However, the naïve notion of equality of linear combinations is finer than equality of the corresponding functions, so we must combine Definition 2 with a definition of equality:
Arguably, even this is not really the correct notion of equality, since functions may be equal for the purpose of integration without being literally equal. If is equipped with a -ideal of null sets (or a -filter of full sets), then we may consider a yet coarser notion of equality:
Sometimes, we wish to restrict attention to those simple functions which we expect to have a finite integral. If is equipped with an ideal of bounded sets (which in a measure space are sets with finite measure), then we may do this:
A simple function of bounded support is a simple function in the sense of Definition 1 such that the fibre over every non-zero number is bounded, or equivalently (in the sense of Definition 2) a formal linear combination of bounded measurable sets.
In some approaches to measure theory, one starts with a -ring of measurable sets, which may be reinterpreted as the bounded sets in the generated -algebra of relatively measurable sets, and then the simple functions will automatically have bounded support.
Finally, there is one more useful restriction (and slight generalisation) of simple functions, applicable when is ordered:
A positive simple function is a simple function in the sense of Definition 1 whose range is contained in the positive cone of , or equivalently (in the sense of Definition 2) a formal -linear combination of measurable sets. An extended positive simple function (note the red herring) takes values in the extended positive cone , or equivalently is a -linear combination.
If is a simple function from to , then we wish to define the integral of . In general, this is a little tricky, but it's easy if either is positive or has bounded support. It is easiest to write down the definition if we think of simple functions using Definition 2. Then we have:
The integral of the simple function , represented by the linear combination , is .
The integral of a positive simple function always exists (but may be infinite). It is finite if is a finite measure, and it is positive (possibly or ) if is a positive measure. Also, if is positive, then the integral of an extended positive simple function always exists.
(However, the integral of an extended positive simple function with respect to a finite positive measure need not be finite.)
The integral of a simple function with bounded support always exists and is finite (being a finite linear combination of finite numbers).
Two (positive or with bounded support) simple functions and are almost equal (with respect to ) if and only if the integral of is zero.
The -norm of a simple function is the integral of its pointwise norm (which is a positive simple function to ) with respect to the absolute value of the measure (which is a positive measure):
In this context, we usually start with a positive measure ; in that case, of course, there is no need to bother taking the absolute value of .
If we don't use almost equality, then we get in general only a seminorm, but if we pass to a quotient space with a norm, then Proposition 3 tells us that we are now using almost equality (and shows that Definition 7 is well defined when applied to Definition 1).
Taking the integral of a simple function of bounded support is a continuous linear functional on , so it extends to all of .
In this way, we may define the integral of any absolutely integrable function.
There might be some technical requirements for this to be true. I'll try to check on that.