I'm terribly sorry about the vague title for this question but I couldn't find a better way to phrase it without writing all of this in the title.
I've been reading in detail the contents of the fifth chapter of Spalsbury & Diestel's 'The Joys of Haar Measure'. I was doing quite well until I reached page 120, where the authors explain a method of assigning an average value to an element of $C(G)$, i.e., the space of continous functions from a compact group $G$ into $\mathbb{R}$.
From my understanding the idea is the following: given a function $f$, if $f$ is constant, then we define that constant value as the average of $f$, which is a perfectly reasonable thing to do. On the other hand, should $f$ not be constant we apply a theorem that guarantees the existence of another function $g$ that gets us 'closer' to a constant function. In more rigorous terminology $g$ is such that Osc$(g)<$Osc$(f)$, where Osc denotes the oscillation of a function.
The function $g$ has an explicit description in terms of $f$, that is $g=RAve_{F}f$, for some finite subset of $G$, where $(RAve_{F}f)(x)=\frac{1}{\mid F\mid}\sum_{a\in F}f(xa)$.
Then, tha authors consider two cases: either g is constant or it isn't. If it is constant then we define that constant as the average of f. If it isn't we apply the theorem again. So far so good.
The first problem arises when we try to inductively generalize this idea. The way I see it, the recursion should be as follows:
$f_{0}=f$, $f_{n+1}=RAve_{F_{n+1}}f_{n}$
whenever we are forced to apply the theorem.
One can easily check that with this definition
Osc$(f_{n+1})<$Osc$(f_{n})$
since we constructed this sequence by repeated application of the theorem mentioned above.
However, the authors claim that this holds:
Osc$(RAve_{F_{n+1}}RAve_{F_{n}}f)<$Osc$(RAve_{F_{n}}f)$
As far as I can tell, the above inequality is saying something completely different. I was wondering if this could be a mistake that was overlooked.
Anyway, after that, the authors show that this sequence has a uniformly convergent subsequence. This is when the second problem arises. The authors seem to completely ignore this observation and don't use it at all in what follows.
In summary, I would like to know what the point was. It could be no more than an interesting fact, but the way it was emphasized makes me think otherwise.
I understand that mine is an extremely specific question that requires familiarity with the book, but I would highly appreciate any insight that you can give me.
If it's of any help at all here's the actual text from the book:
