Let $A$ be the space of functions from $[0,1]$ to $\mathbb{R}$. Is there a well-defined operator, say $p$, that gives the size of $f\in A$'s range? For example, if $f(x)=\begin{cases}0&\text{ if }x\leq 0.5\\1&\text{ otherwise}\end{cases}$, then $p(f)=2$.
Is there a such operator that is easy to work with, i.e. $p(f)=\int_0^1 f(x)w(x)dx$ for some $w(x)$? (I think this would not be an answer, but just wanted to emphasize that I want the operator to be easy to work with).
If we don't assume anything about $A$ like it being a space of continuous functions, then your question doesn't make sense. If we make such an assumption, then one way would be $$p(f)=2\min_{t\in \Bbb R}\left(\lim_{n\to\infty}\sqrt[n]{\int_0^1|f(x)-t|^ndx}\right)$$in other words, move $f$ up or down until the sup norm is at its minimum, and then double that sup norm.