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abstract: 'A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semi-processes in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.'
author:
- 'Jorge E. Cardona and Lev Kapitanski'
title: 'Measurable process selection theorem and non-autonomous inclusions'
---

Introduction
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Let \[aut\] = f(u) be an archetypical autonomous differential equation. Autonomous refers to the structure of the equation and means that the independent variable, $t$, does not appear explicitly (independently) in the equation. Because of that, is invariant under the time shift (translation) $\theta_\tau: t\mapsto t+\tau$, and if $u(\cdot)$ is a solution of , then $\theta_\tau u(\cdot) = u(\cdot + \tau)$ is a solution as well. Suppose describes evolution/dynamics on some set $X$ (which could be a finite- or infinite-dimensional vector space or manifold). Given an $a\in X$, let $u(t, a)$, $t\in [0, +\infty)$ be a solution of starting at $u(0, a) = a$ (let us assume that global solutions exist forward in time). If $v(\cdot, u(t_1, a))$ is a solution of starting (at $t = 0$) from the point $u(t_1, 