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 .
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June 14, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
unapologetic.wordpress.com’s done it once again. Amazing article.
Hi,
That’s great: I was exactly looking for that result somewhere (googling ‘mean value theorem measure’, you came up second – I am rarely so lucky…). Do you know a reference (text book/article) that I could refer to to mention this result?
Thanks a lot,
Armando
I believe it shows up in Halmos’ Measure Theory.
Perfect, there it is, thanks again
Armando
No problem. Remember to tell your friends where to look