Extending the Integral

Given an integrable function , we’ve defined the indefinite integral to be the set function

This is clearly real-valued, and we’ve seen that it’s countably additive. If is a.e. non-negative, then will also be non-negative, and so the indefinite integral is a measure. Since is integrable we see that

and so is a totally finite measure.

But this situation feels a bit artificially restrictive in a couple ways. First of all, measures can be extended real-valued — why do we never find ? Well, it makes sense to extend the definition of at least the symbol of integration a bit. If is not integrable, but a.e., there is really only one possibility: there is no upper bound on the integrals of simple functions smaller than . And so in this situation it makes sense to define

Similarly, if a.e. and fails to be integrable, it makes sense to define

In general, we can break a function into its positive and negative parts and , and then define

for all functions for which at most one of and fails to be integrable. That is, if the positive part is integrable but the negative part is not, then the integral can be defined to be . If the negative part is integrable but the positive part isn’t, we can define the integral to be . If both positive and negative parts are integrable then the whole function is integrable, while if neither part is integrable we still leave the integral undefined. We don’t know in general how to deal with the indeterminate form .

And so now we find that any a.e. non-negative function — integrable or not — defines a measure by its indefinite integral. If isn’t integrable, then we get an extended real-valued set function, but this doesn’t prevent it from being a measure. As a matter of terminology, we should point out that we don’t call a function whose integral is now defined to be positive or negative to be “integrable”. That term is still reserved for those functions whose indefinite integrals are totally finite, as above.