I can try to define an averaging operator for functions, namely let $$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$ by $$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$ whenever the limit converges. My question is: is there any intuitive description of the domain $D$ (other than the above limit converging)? A few observations:

- Clearly if $f$ is compactly supported then $Af = 0$, in particular $f \in D$.
- There exist non-compactly supported $f \in D$. E.g. constant functions, $sin$, $cos$, and many others.
- There
*are*bounded functions which are*not*in $D$. Maybe there is a more concise example but here is an intuitive one: $f$ will take either the value 0 or 3. It should start say as 0 on [0,1], then it will be 3 for long enough to bring the average up to 2, then it will return to 0 for long enough to bring the average down to 1, and so on. Thus the average will fluctuate between $1$ and $2$ and never converge.

Thoughts?

closureof $D$? – Nate Eldredge May 13 '12 at 14:39