  The main subject of this paper is the analysis of sequences of exponentially
bounded integrated semigroups which are related to Cauchy problems
\begin{equation}\label{jed} \frac{\partial}{\partial
t}u(t,x)-a(D)u(t,x)=f(t,x), \quad u(0,x)=u_0(x), \quad t\geq 0, \ x\in \mathbb
R^d, \end{equation} with a distributional initial data $u_0$ and a
distributional right hand side $f$ through a sequence of equations with
regularized $u_0$ and $f$ and a sequence of (pseudo) differential operators
$a_n(D)$ instead of $a(D)$. Comparison of sequences of infinitesimal generators
and the determination of corresponding sequences of integrated semigroups are
the main subject of the paper. For this purpose, we introduce association, the
relation of equivalence for infinitesimal generators on one side and the
corresponding relations of equivalence of integrated semigroups on another
side. The order of involved assumptions on generators essentially characterize
the mutual dependence of sequences of infinitesimal generators and the
corresponding sequences of integrated semigroups.
