Mean value theorems for the Haar integral? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:01:58Z http://mathoverflow.net/feeds/question/120121 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120121/mean-value-theorems-for-the-haar-integral Mean value theorems for the Haar integral? Alex M 2013-01-28T16:08:22Z 2013-01-28T18:06:30Z <p>Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?</p> <p>In general, are there mean value theorems for abstract spaces with measures? (Or at least for Borel measures?)</p> <p>Later edit: After reading the first two comments, let me be more precise; I'm looking for a theorem giving something like: for any continuous $f$ on $G$, there exist $x \in G$ such that $\int_G f(g) \mathrm{d}g = f(x)$. Does such an $x$ really exist? Can anything else be said about it (the integral being so special, maybe this $x$ can be made more precise)?</p> http://mathoverflow.net/questions/120121/mean-value-theorems-for-the-haar-integral/120132#120132 Answer by Gerald Edgar for Mean value theorems for the Haar integral? Gerald Edgar 2013-01-28T18:06:30Z 2013-01-28T18:06:30Z <p>Say $\mu$ is a Borel probability measure on a connected set $A$ in a topological space. Let $f : A \to \mathbb R$ be continuous. Then the mean value $\int_A f\;d\mu$ is equal to $f(a)$ for some $a \in A$. Proof: the mean value is between the sup of all values and the inf of all values, so (by connectedness) it is a value of the function. </p> <p>Of course the desired result fails for non-connected sets. Even in the two-point group we get a counterexample.</p>