If you want better than $\aleph_0$ additivity, then you probably don't want values in the reals. There is no non-trivial convergent series of reals with uncountably many nonzero terms.

Another topic: A $\sigma$-smooth linear funtional $L$ on $C(X)$, the set of continuous real-valued functions on a topological space $X$, is a linear functional such that: if $f_n$ is a sequence that decreases pointwise to $0$, then $L(f_n) \to 0$. A generalization is $\tau$-smooth linear functional: if $\mathcal A$ is a family of nonnegative functions in $C(X)$ that is directed in the sense: for any $f_1, f_2 \in \mathcal A$ there exists $f_3 \in \mathcal A$ with $f_3 \le f_1$ and $f_3 \le f_2$, then $\inf_{f \in \mathcal A} L(f) = 0$. For some topological spaces, every $\sigma$-smooth functional is $\tau$-smooth. For other topological spaces, the two concepts are different.

A related topic: A $\sigma$-smooth linear funtional $L$ on $C(X)$, the set of continuous real-valued functions on a topological space $X$, is a linear functional such that: if $f_n$ is a sequence that decreases pointwise to $0$, then $L(f_n) \to 0$. A generalization is $\tau$-smooth linear functional: if $\mathcal A$ is a family of nonnegative functions in $C(X)$ that is directed in the sense: for any $f_1, f_2 \in \mathcal A$ there exists $f_3 \in \mathcal A$ with $f_3 \le f_1$ and $f_3 \le f_2$, then $\inf_{f \in \mathcal A} L(f) = 0$. For some topological spaces, every $\sigma$-smooth functional is $\tau$-smooth. For other topological spaces, the two concepts are different.