Lebesgue discovered that a satisfactory theory of integration results if the sets $E_i$ are allowed to belong to a larger class of subsets of the line.
Two consecutive elements do not belong both to $A$ or both to $B$.
It turns out that $A$, $B$ and $C$ all belong to the same class, which we represent by the symbol $P_2$.
For the sake of clarity, we shall indicate in what follows to which space $X$ belongs.
[Do not write: “$E$ belongs to the most important classes of algebras” if you mean: $E$ is one of the most important classes of algebras or $E$ is among the most important classes of algebras.]