Now we’ve got a definition of the integral of a wider variety of functions than before, so let’s look over the basic properties.
First of all, from what we know about convergence in measure and algebraic and order properties of integrals of simple functions, we can see that if and are integrable functions and is a real number, then so are the absolute value , the scalar multiple , and the sum . As special cases, we can see that the positive and negative parts
are both integrable.
If is a measurable set and is a mean Cauchy sequence of integrable simple functions converging in measure to , then it should be clear that is mean Cauchy and converges in measure to . Thus we can define
Since integrals of simple functions are linear, and limits are linear, we immediately conclude that
as before, but now for general integrable functions. Similarly, if is nonnegative a.e., we can find a sequence of nonnegative simple functions converging a.e. (and thus in measure) to . The integral of each of these functions is nonnegative, and so their limit must be as well. That is, if a.e., then
Now, all our properties that we proved using only these two linearity and order properties follow. If a.e., then
For any two integrable functions amd we find
If and are real numbers so that for almost all , then we have
and if an integrable function is a.e. nonnegative, then its indefinite integral is monotone.
It takes a bit of work, though, to check that the indefinite integral of an integrable function is absolutely continuous. If is a mean Cauchy sequence converging in measure to , then we have
We know that the indefinite integrals of the are uniformly absolutely continuous, so we have control over the size of the first term on the right. The second term on the right can be kept small by choosing a large enough , since converges in measure to .
Finally, the indefinite integral of is countably additive; since is a mean Cauchy sequence of simple functions we can write as the limit , and we know that this limit is countably additive.