Basic Properties of Integrable Simple Functions
First of all, if is simple then the scalar multiple is simple for any real number . Indeed, , and so the exact same sets must have finite measure for both and to be integrable. It’s similarly easy to see that if and are both integrable, then is integrable. Thus the integrable simple functions form a linear subspace of the vector space of all simple functions.
Now if is integrable then the product is integrable whether or not is. We can write
If each has finite measure, then so does each , whether each does or not. Thus we see that the integrable simple functions form an ideal of the algebra of all simple functions.
We can use this to define the integral of a function over some range other than all of . If is an integrable simple function and is a measurable set, then is again an integrable simple function. We define the integral of over as
This has the effect of leaving the same on and zeroing it out away from . Thus the integral over the rest of the space contributes nothing.
The next two properties are easy to prove for integrable simple functions, but they’re powerful. Other properties of integration will be proven in terms of these properties, and so when we widen the class of functions under consideration we’ll just have to reprove these two. The ones we will soon consider will immediately have proofs parallel to those for simple functions.
Not only is the function integrable, but we know its integral:
Indeed, if you were paying attention yesterday you’d have noticed that we said we wanted integration to be linear, but we never really showed that it was. But it’s not really complicated: the expression represents as a simple function, and it’s clear that the formula holds as stated.
Almost as obvious is the fact that if is nonnegative a.e., then . Indeed in any representation of as a simple function, any term corresponding to a negative must have or else wouldn’t be nonnegative almost everywhere. But then the term contributes nothing to the integral! Every other term has a nonnegative and a nonnegative measure , and thus every term in the integral is nonnegative. This is the basis for all the nice order properties we will find for the integral.