On stability of solutions of integral equations in the class of measurable functions

Consider the equation $G(x)=\tilde{y},$ where the mapping $G$ acts from a metric space $X$ into a space $Y,$ on which a distance is defined,
$\tilde{y} \in Y.$ The metric in $X$ and the distance in $Y$ can take on the value $\infty,$ the distance satisfies only one property of a metric:
the distance between $y, z \in Y$ is zero if and only if $y=z.$ For mappings $X \to Y$ the notions of sets of covering, Lipschitz property, and closedness are defined.
In these terms, the assertion is obtained about the stability in the metric space $X$ of solutions of the considered equation to changes of the mapping $G$ and the element
$\tilde{y}.$ This assertion is applied to the study of the integral equation
 $f(t,{\int }_0^{1}K(t,s\left)x\right(s)ds,x(t\left)\right)=y~\left(t\right),t\in [0,1],$
with respect to an unknown Lebesgue measurable function $x: [0,1] \to \mathbb {R}.$ Sufficient conditions are obtained for
the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions $f, \mathcal{K}, \tilde{y}.$