We deal with the implicit integral equation $$h\left(u\left(t\right)\right)=f(t,{\int }_{I}g(t,z)u\left(z\right)dz)\text{for}\text{a.a.}t\in I,$$ where $I:=[0,1]$ and where $f:I\times [0,\lambda ]\to ℝ$, $g:I\times I\to [0,+\infty [$ and $h:]0,+\infty [\to ℝ$. We prove an existence theorem for solutions $u\in L^s\left(I\right)$ where the contituity of $f$ with respect to the second variable is not assumed.

Suppose that $(X,𝒜)$ is a measurable space and $Y$ is a metrizable, Souslin space. Let ${𝒜}^u$ denote the universal completion of $𝒜$. For $x\in X$, let $\underline{f}(x,·)$ be the lower semicontinuous hull of $f(x,·)$. If $f:X\times Y\to \overline{ℝ}$ is $({𝒜}^u\otimes ℬ\left(Y\right),ℬ\left(\overline{ℝ}\right))$-measurable, then $\underline{f}$ is $({𝒜}^u\otimes ℬ\left(Y\right),ℬ\left(\overline{ℝ}\right))$-measurable.