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A part of the example is given in the following picture:

enter image description here

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?

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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.

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