How to interpret composition of Integral operators

Question

Let $Tf(y):=\int_A k_T(x,y)f(x)dx$ and similarly for $S$ with kernel $k_S$.

How do I interpret $STf(y)$?

The book I'm reading states that $k_{ST}(x,z):=\int k_S(y,z)k_T(x,y)dy$. Why is that?

Answer 1

$$\begin{align} STf(z) = S(Tf)(z) &= \int k_S(y,z) \, Tf(y) \, dy \\ &= \int k_S(y,z) \, \left( \int k_T(x,y) \, f(x) \, dx \right) \, dy \\ &= \iint k_S(y,z) \, k_T(x,y) \, f(x) \, dx \, dy \\ &= \int \left( \int k_S(y,z) \, k_T(x,y) \, dy \right) f(x) \, dx \\ &=: \int k_{ST}(x,z) \, f(x) \, dx \end{align}$$