Question about Banach Limit I just studied the Banach Limit $\Lambda$ in my Functional Analysis class and I am trying to answer the following question

Can one define a generalised limit for bounded functions $f : \mathbb{R} \to \mathbb{R}$?

For now, I constructed a functional $\mathcal{L}$ from the space $\mathcal{B}$ of bounded linear functions to $\mathbb{R}$ in the following way:

*

*notice that every real sequence $(x_n)_n$ can be seen as a function $f: \mathbb{N} \to \mathbb{R}$;

*notice that the set $S=\{s \text{ such that } s:\mathbb{N} \to \mathbb{R}, s \text{ bounded}\}$ forms a subspace of $\mathcal{B}$;

*use Hahn-Banach Theorem to define a linear functional $\mathcal{L}$ extending $\Lambda$ from $S$ to all $\mathcal{B}$ and such that
$$
|\mathcal{L}(f)| \leq \|f\| \quad \text{ for every } f \in \mathcal{B}.
$$
Such functional is linear and normalized by construction, and positivity is pretty straightforward. I am having trouble with shift invariance. Any suggestions?
 A: I guess $S$ is a quotient of $\mathcal{B}$,  rather than a subspace,  correct?
That is, we may define a projection mapping
$\pi :\mathcal{B}\to S, $
by letting $\pi (f)$ be the restriction of $f$ to $\mathbb N$, for every $f$ in $\mathcal B$.
Choosing a Banach limit $\Lambda :S\to \mathbb R$, we  then define  $\mathcal{L}$ to be the composition
$$
  \mathcal B \quad {\buildrel \pi \over \longrightarrow} \quad S \quad {\buildrel \Lambda \over \longrightarrow} \quad \mathbb R.
  $$
The OP has already observed  that this functional has all of the required properties, except possibly for translation invariance.
Denoting by $T_y(f)$ the translation of $f$ by $y$,  namely
$$
  T_y(f)|_x =  f(x-y),
  $$
the missing property is thus
$$
  \mathcal L(T_y(f)) = \mathcal L(f).
  \tag 1
  $$
Observe however  that $\Lambda $ is invariant by integer translations, so this  easily implies that
(1) holds provided $y$ is an integer number.  In particular the correspondence
$$
  y\mapsto \mathcal L(T_y(f))
  \tag 2
  $$
is seen to be periodic with period 1.
When desired invariance properties are absent,
it is a common trick in the theory of group representations to average out.  That is,
in order to overcome the above difficulty
let us   define   $\tilde{\mathcal L}$ by
$$
  \tilde{\mathcal L}(f) = \int_0^1 \mathcal L(T_y(f))\, dy.
  $$
Given any $z$ in $\mathbb R$, we then have that
$$
  \tilde{\mathcal L}(T_z(f)) =
  \int_0^1 \mathcal L(T_{y+z}(f))\, dy =
  \int_z^{z+1} \mathcal L(T_{y}(f))\, dy =
  \int_0^1 \mathcal L(T_{y}(f))\, dy = \tilde{\mathcal L}(f),
  $$
where the penultimate  step is due to the periodicity of (2).
It is now easy to see that $\tilde{\mathcal L}$ satisfies all of the required conditions.
