Is there a general definition of Maximal function? I see this term coming up quite a lot, but I have not figured if there's a general definition.
For example wikipedia https://en.wikipedia.org/wiki/Maximal_function shows several examples but does not provide a general definition (if there's any).
In my understanding a maximal function is any function involving a $ \sup $. For example in the proof of Caratheodory theorem, derivatives of measures or Riesz representation theorem they all involve supremum of something.
Is there a more general definition rather than per example based?

A good definition of a maximal function would be a function using a supremum. Since maximal and maximize have the same root word, I think this is what people mean by "maximal function."
