0

In my Real Analysis class I've often seen the Upper Riemann Sum (in the definition of Riemann integrability) defined as: $$\sup_P\{U(f,P)\}$$ This seems to mean the supremum of all possible upper sums (corresponding to all possible partitions of the interval $[a,b]$. Of course, what we're changing to get the elements of this set is the partition. My question is are there other instances where the "supremum with respect to" notation is used? And am I thinking of this correctly as $P$ being a parameter here, the same way $i$ is a parameter in a sum?

iceburger
  • 101
  • 2
    Yeah I think of $P$ as a parameter since the partition can be rather arbitrary. You'll see other situations where the set will be a subscript as a parameter say in a sum, but I feel like in analysis everyone just makes up their own notation a lot of the time. – CyclotomicField May 02 '22 at 00:58
  • This is kind of stuff is used all the time. For a sample of similar ideas in the context of measure theory, peruse this: https://math.stackexchange.com/questions/3385011/definitions-of-measurability-outer-inner-measure-convergence-vs-caratheodory-c – B. S. Thomson May 02 '22 at 03:13
  • If $\mathcal R[x]$ is some logical condition on the variable $x$, and $f$ is some function defined at least on all $x$ for which $\mathcal R[x]$ holds, then the notation $\sup_{\mathcal R[x]} f(x)$ is defined by $$\sup_{\mathcal R[x]} f(x) = \sup {f(x) \mid \mathcal R[x]}$$ Your notation is a shorthand for this, where the condition "is a partition of $[a, b]$" is understood. It should not include the set brackets as it does, since that part of the notation does not by itself define the appropriate set. But someone tossed them in anyway since suprema are a property of sets. – Paul Sinclair May 02 '22 at 16:27

0 Answers0