$c=$The set of all real convergent sequences
$l_\infty=$The set of all real bounded sequences
Clearly $c\subset l_\infty$
$f:c\to \mathbb R$ is called limit functional defined by $f(x)=\lim\limits_{n\to\infty} x_n$
There exist infinitely many extensions of $f$ to $l_\infty$. From here main concept of Banach Limit functional comes.
With the help of this Banach limits the notion of almost convergence is defined.
My Question : What happens if we take extensions of the statistical limit functional $g$?
(where $g:st\to \mathbb R$ is defined by $g(x)=stat\lim\limits_{n\to\infty} x_n$)
Here, $st=$ The set of all bounded statistically convergent sequences.
May we define a new kind of convergence in this approach? Is it not available in literature?