The Unapologetic Mathematician

Mathematics for the interested outsider

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 f is an essentially bounded measurable function with \alpha\leq f\leq\beta a.e. for some real numbers \alpha and \beta, and if g is any integrable function, then there is some real number \gamma with \alpha\leq\gamma\leq\beta so that

\displaystyle\int f\lvert g\rvert\,d\mu=\gamma\int\lvert g\rvert\,d\mu

Actually, this is a statement about finite measure spaces; the function g is here so that the indefinite integral of \lvert g\rvert will give us a finite measure on the measurable space X to replace the (possibly non-finite) measure \mu. This explains the g 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 \alpha\leq f\leq\beta a.e., and so \alpha\lvert g\rvert\leq f\lvert g\rvert\leq\beta\lvert g\rvert a.e. as well. This tells us that f\lvert g\rvert is integrable. And thus we conclude

\displaystyle\alpha\int\lvert g\vert\,d\mu\leq\int f\lvert g\rvert\,d\mu\leq\beta\int\lvert g\rvert\,d\mu

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

\displaystyle\alpha\leq\frac{\int f\lvert g\rvert\,d\mu}{\int\lvert g\vert\,d\mu}\leq\beta

this term in the middle is our \gamma.

June 14, 2010 Posted by | Analysis, Measure Theory | 5 Comments