Functional analytic approach to Ces\`aro mean

We study a certain class $\mathcal{P}$ of positive linear functionals $\varphi$ on $L^{\infty}([1,\infty))$ for which $\varphi(f) = \alpha$ if $\lim_{x \to \infty} \frac{1}{x} \int_1^x f(t)dt = \alpha$. It turns out that translations $f(x) \mapsto f(rx)$ on $L^{\infty}([1, \infty))$, where $r \in [1, \infty)$, which are induced by the action of the multiplicative semigroup $[1, \infty)$ on itself, plays an intrinsic role in the study of $\mathcal{P}$. We also deal with an analogue $\mathcal{K}$ of $\mathcal{P}$ of positive linear functionals on $L^{\infty}([0, \infty))$ partaining to the action of the additive semigroup $[0, \infty)$ on itself. In particular, we give some expressions of maximal possible values of $\mathcal{P}$ and $\mathcal{K}$ for a given function respectively.