Linear and multiplicative limit functional

Question

The popular generalizations for the standard limit functional are the linear Banach limits, and multiplicative ultra filter limits. An extended functional cannot be both as Banach limits also are shift invariant, and the three don’t work together well, but I remember reading in a text that a solely linear and multiplicative limit does exist.

So I ask if there’s any text on or interesting properties of a limit with the following properties:

1. $L(ax+by)=aL(x)+bL(y)$
2. $L(xy)=L(x)L(y)$
3. $L(x)=lim(x) \iff \exists lim(x)$
4. $lim inf (x) \le L(c) \le limsup(x) \iff x_i \in \Bbb R$

Like how many are there (there are $2^{2^{\aleph_0}}$ Banach limits), how do they interact with other things in math like continuity, measurability, and how useful would such a limit be in comparison to the other forms of generalized limit?

I assume all such functionals are continuous, but I’m not certain on that either.