Let

$\sigma $ be a mapping of the set of positive integers into itself. A continuous linear functional

$\phi $ on the space

${\ell}^{\infty}$ of real bounded sequences is a

$\sigma $-mean if

$\phi \left(x\right)\ge 0$ when the sequence

$x=\left({x}_{n}\right)$ has

${x}_{n}\ge 0$ for all n,

$\phi \left(e\right)=1$ where

$e:=(1,1,\xb7\xb7\xb7)$, and

$\phi \left(\left({x}_{\sigma \left(n\right)}\right)\right)=\phi \left(x\right)$ for all

$x\in {\ell}^{\infty}$. Let

${V}_{\sigma}$ be the space of bounded sequences all of whose

$\sigma $-means are equal, and let

$\sigma $-lim x be the common value of all

$\sigma $-means on x. In the special case in which

$\sigma \left(n\right):=n+1$ the

$\sigma $-means are exactly the Banach-limits, and

${V}_{\sigma}$ is the space of all almost convergent sequences considered by

*G. G. Lorentz* [Acta Math. 80, 167-190 (1948;

Zbl 0031.29501)]. In a natural way the author of this paper introduces the space

$B{V}_{\sigma}$ of sequences of

$\sigma $-bounded variation, which is a Banach space. Then he characterizes all real infinite matrices A, which are absolutely

$\sigma $-conservative (absolute

$\sigma $-regular). Thereby A is said to be absolutely

$\sigma $-conservative if and only if

$Ax\in B{V}_{\sigma}$ for all

$x\in bv$, where bv denotes the space of sequences of bounded variation, and A is said to be absolutely

$\sigma $-regular if and only if A is absolutely

$\sigma $-conservative and

$\sigma -limAx=limx$ for all

$x\in bv$.