This paper proves the existence of solutions to the initial value problem
\[
(
I
V
P
)
{
x
′
(
t
)
=
f
(
t
,
x
(
t
)
)
(
0
≤
t
≤
1
)
,
x
(
0
)
=
0
,
(\mathrm {IVP})\qquad \qquad \left \{\begin {array}{l} x’(t)=f(t,x(t))\qquad \quad (0\le t\le 1), x(0)=0,\end {array} \right .
\]
where
f
:
[
0
,
1
]
×
R
M
→
R
M
f:[0,1]\times \mathbb {R}^M\to \mathbb {R}^M
may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set
M
M
can be arbitrarily large (finite or infinite); our theorem is new even for
card
(
M
)
=
2
\mbox {card}(M)=2
. The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.