When trying to determine whether a function $f$ is Riemann integrable or not, we're typically in the following situation:
- Write $U_f(P)$ and $L_f(P)$ for the corresponding "upper Riemann sum" and "lower Riemann sum" with respect to a partition $P$.
- Hence $\mathrm{img}(U_f)$ and $\mathrm{img}(L_f)$ denote the set of upper sums and the set of lower sums respectively.
- Whenever $B$ and $A$ are subsets of the real line, write $B \geq A$ to mean the following: $$\mathop{\forall}_{b \in B, a \in A}b \geq a$$
- We know that $\mathrm{img}(U_f) \geq \mathrm{img}(L_f).$
- We want to know whether or not $\mathrm{inf}(\mathrm{img}(U_f)) = \mathrm{sup}(\mathrm{img}(L_f))$, i.e. whether or not these sets are "touching"; if they are, then the Riemann integral of $f$ exists, and if not, then not.
To reduce my overall feelings of confusion about this situation, I decided to try "doing maths" on it, which for me usually involves coming up with definitions, notation and terminology to help "conceptualize around" the situation. Here's what I came up with.
Whenever $P$ is a poset, write $\overline{P}$ for the collection of pairs $(B,A)$ of non-empty subsets of $C$ satisfying $B \geq A$.
Also, call $(B,A) \in \overline{P}$ touching iff there is at most one $p \in P$ with $B \geq p \geq A$. (By this inequality, I mean what you think I mean.)
Now assuming $P$ is conditionally complete, we have functions $$\overline{\inf}, \overline{\sup}:\overline{P} \rightarrow P$$ given as follows:
$$\overline{\inf}(B,A) = \mathrm{inf}(A), \qquad \overline{\sup}(B,A) = \mathrm{sup}(A)$$
Its clear that $\overline{\inf} \geq \overline{\sup}$ holds pointwise; furthermore, it can be seen that being a touching element of $\overline{P}$ is equivalent to being an element of the equalizer of $\overline{\inf}$ and $\overline{\sup}$.
There's also an obvious way to order $\overline{P}$; we find that $\overline{\inf}$ becomes antitone and $\overline{\sup}$ becomes monotone. With appropriate tweaks to this, we can probably show that the touching pairs are precisely the maximal elements of $\overline{P}$. I haven't thought about the details.
We can also show that if $P$ equals $\mathbb{R}$ or $\mathbb{Q}$, then $(B,A) \in \overline{P}$ is touching iff for all $\varepsilon \in P,$ there exists $b \in B$ and $a \in A$ such that $\varepsilon \geq b-a.$ A reasonable question to ask is: how far can this be generalized?
Anyway, what I'd like to know is:
Question. Is there standard terminology or notation for any or all of the above concepts or variants thereon?