This note is concerned with the local existence of solutions to the initial value problem (*)

${x}^{\text{'}}=f(t,x)$,

$x\left(0\right)=\theta $, where

$f$ maps

$[0,a]\times B$ into a Banach space E and B is a ball in E centered at the origin. It is assumed that

$t\to f(t,x)$ is measurable,

$x\to f(t,x)$ is continuous, and

$\parallel f(t,x)\parallel \le m\left(t\right)$ for

$(t,x)\in [0,a]\times B$ where

${\int}_{0}^{a}m\left(s\right)ds<\infty $. The main assumption on f involves the Kuratowski (or ball) measure of noncompactness and has the form that for each

$\u03f5>0$ and

$X\subset B$, there is a closed subset

${I}_{\u03f5}$ of [0,a] such that

$\mu [I-{I}_{\u03f5}]<\u03f5$ (

$\mu $ denotes Lebesgue measure) and

$\alpha \left(f(T\times X)\right)\le {sup}_{t\in T}h(t,\alpha \left(X\right))$ for each compact subset T of

${I}_{\u03f5}$ (It seems that there is a misprint in the statement of this theorem since it is stated in the paper that T is a compact subset of I rather than

${I}_{\u03f5})$. The function h satisfies the Caratheodory conditions on

$[0,a]\times [0,\infty )$ and

$u\left(t\right)\equiv 0$ is the maximal solution to

${u}^{\text{'}}=h(t,u)$,

$u\left(0\right)=0$. Under these hypotheses it is shown that the initial value problem (*) has a solution. The result is an improvement of a preceding one of

*G. Pianigiani* [ibid. 23, 853-857 (1975;

Zbl 0317.34050)].