Consider the scalar i.v.p. in ${\mathbb R}$ $$ x'=f(t,x), \; t\in[0,T], \; x(0)=x_0, $$ where $T\in {\mathbb R}$, $T>0$, $x_0\in {\mathbb R} $, and $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$ has the property:
(i) for each (Lebesgue) measurable $y: [0,T] \mapsto {\mathbb R}$, the map $[0,T]\ni t\mapsto f(t,y(t))\in {\mathbb R}$ is measurable.
(ii) For almost all $t\in[0,T]$, $\sup_{x\in R}|f(t, x)|\leq l(t)$, where $l : [0, T] \mapsto R$ is Lebesgue integrable.
I am aware of (several) results showing the existence of Carath'{e}odory solutions to the above problem, in the case when $f$ is not necessarily continuous, but, in addition to (possibly stronger versions) of (i), and (ii), one supposes that $f$ is non-decreasing in some sense (see. eg., https://projecteuclid.org/euclid.die/1368638179; https://doi.org/10.1090/S0002-9939-97-03942-7, etc ).
QUESTION: is there any similar result proving the existence of Carath'{e}odory solutions to the above problem in the case when $f$ is discontinuous, but non-increasing?
