The Integral Mean Value Theorem

We have an analogue of the integral mean value theorem that holds not just for single integrals, not just for multiple integrals, but for integrals over any measure space.

If is an essentially bounded measurable function with a.e. for some real numbers and , and if is any integrable function, then there is some real number with so that

Actually, this is a statement about finite measure spaces; the function is here so that the indefinite integral of will give us a finite measure on the measurable space to replace the (possibly non-finite) measure . This explains the in the multivariable case, which wasn’t necessary when we were just integrating over a finite interval in the one-variable case.

Okay, we know that a.e., and so a.e. as well. This tells us that is integrable. And thus we conclude

Now either the integral of is zero or it’s not. If it’s zero, then is zero a.e., and so is , and our assertion follows for any we like. On the other hand, if it’s not we can divide through to find

this term in the middle is our .