Can someone clarify in simple terms what it means to "apply an inequality to a measure"?

Question

I am reading wikipedia's entry on Holder's inequality https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality#Counting_measure

There are quite a few variations of Holder's inequality "applied to different measures" - a phrase quite new to me. I have no idea what this notion of a measure is, when I look up, for example, the counting measure, it says that the counting measure $\nu: S \to [0, \infty]$, is the size of a set $A$. https://www.ma.utexas.edu/users/gordanz/notes/measures.pdf

Can someone please clarify what it means to "apply an inequality to different measures". I am not a mathematician, what is a simple way to understand this idea? Must we specify which measure we are applying an inequality against?

Answer 1

Hölder's inequality is an inequality involving integrals. You may be familiar with ordinary integrals of functions $f:\mathbb{R}\to\mathbb{R}$, but integrals can be defined in much more general contexts as well. A "measure" on a set $S$ is a gadget that allows you to define integrals of functions $f:S\to\mathbb{R}$ (the measure on $\mathbb{R}$ that gives ordinary integration is called "Lebesgue measure"). Hölder's inequality is valid when talking about the integrals obtained by any measure in this way. So "applying Hölder's inequality to a measure" just means using Hölder's inequality where the integrals in the inequality are integrals defined using that measure.

Answer 2

Holder's inequality is a very general result concerning very general integrals in an arbitrary measure space. This is probably the most remarkable thing about Holder's inequality, and why it is so useful.

Even in the statement of the theorem (as stated in the wikipedia link) the result only requires that we have a measure space $(S,\Sigma,\mu)$. All they mean by "applied to different measures" is that we can explore some meaningful implications and say explicitly what the inequality means if we are given a priori certain popular measure spaces (e.g., counting measure, Lebesgue measure, etc...).