A part of the example is given in the following picture:
But I did not understand why for each $t$, $k(t,s)f(s)$ is Lebesgue measurable on [a,b], could anyone explain this for me please?
For fixed $t$ let $g$ be defined by $g(s)=k(t,s)$. Since $k$ is measurable, $g$ is measurable.
We have $k(t,s)f(s)=g(s)f(s)$ and the product of measurable functions is measurable.