A positive functional x∗ on the space ℓ∞ of all bounded sequences is called a Banach–Mazur limit if ‖x∗‖=1 and x∗x=x∗Tx for all x=(x1,x2,…)∈ℓ∞, where T is the forward shift operator on ℓ∞, i.e., Tx=(0,x1,x2,…). The set of all Banach–Mazur limits is denoted by BM and a collection of extreme points of BM is denoted by extBM. Let ac0={x∈ℓ∞:x∗x=0for allx∗∈BM}. The following sequence spaces D(ac0)={x∈ℓ∞:x⋅ac0⊆ac0}andI(ac0)=ac0+−ac0+ are studied. In particular, if z∈ℓ∞ then z∈D(ac0) iff z−Tz∈I(ac0); moreover, z∈D(ac0) iff x∗{n:|zn−x∗z|≥ϵ}=0 for all ϵ>0 and x∗∈extBM. Order properties of Banach–Mazur limits are considered. Some properties of extBM are derived. We used the representation of functionals x∗∈BM as Borel measures on βN∖N. The cardinalities of some subset of BM are given. We also consider some questions of the probability theory for finite additive measures. E.g., for every x∗∈BM there exists an element x∈ℓ∞ such that the distribution function Fx∗,x(t)=x∗{n:xn≤t} is continuous on R. Two definitions of a variance are suggested. It is shown that Radon–Nikodym theorem is not valid for finite additive measures: the relations 0≤x∗≤y∗∈ℓ∞∗ do not imply the existence of w∈ℓ∞ satisfying x∗x=y∗(wx) for all x∈ℓ∞.