We prove a global existence result for bounded solutions to a class of abstract semilinear delay evolution equations with measures subjected to nonlocal initial data of the form
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)=\{Au(t)+f(t,u_t)\}\mathrm{d}t+\mathrm{d}g(t),&{}\quad t\in \mathbf{R}_+,\\ u(t)=h(u)(t),&{}\quad t\in [\,-\tau ,0\,], \end{array}\right. \end{aligned}$$
where
$$\tau \ge 0$$
,
$$A:D(A)\subseteq X\rightarrow X$$
is the infinitesimal generator of a
$$C_0$$
-semigroup,
$$f:\mathbf{R}_+\times \mathcal {R} ([\,-\tau ,0\,];X)\rightarrow X$$
is continuous,
$$g\in BV_{\mathrm{loc}}(\mathbf{R}_+;X)$$
, and
$$h:\mathcal {R} _b(\mathbf{R}_+;X)\rightarrow \mathcal {R} ([\,-\tau ,0\,];X)$$
is nonexpansive.