Measurabilty of a function

Question

Let $k:X\times Y\rightarrow \mathbb{R}$ be a measurable function, where $(X,\mu)$ and $(Y,\nu)$ are finite measure spaces. Further let $g: X\rightarrow \mathbb{R}$ be a positive and measurable function. Suppose for each $x\in X$ we can find a non-negative $s$ such that $\int_{Y}(|(k(x,y)|\wedge s\ ) d\nu(y) =g(x)$. Here $a\wedge b$ is the minimum of the values $a$ and $b$. Define now $s(x):=\inf\{s\geq 0| \int_{Y}|k(x,y)|\wedge s=g(x)\} $. Is $s$ measurable?