I have been studying transition kernels recently from "Probability and Stochastics" by Erhan Cinlar. The book shows that for any transition kernel from $(E, \mathcal{E})$ to $(F, \mathcal{F})$, and for any $x \in E$, a measure on $(F, \mathcal{F})$ is defined which will be henceforth be signed by $K(x)$.
Kernels can therefore be naturally thought of as mappings: $$ K: \mathcal{F}_+ \rightarrow \mathcal{E}_+ $$ by: $$ \left(Kf\right)(x)=\int_F f(y)\mathrm{d}K(x) $$
Later, the book expands this definition by defining the following mapping for an s-finite kernel $K$: $$ \tau: (\mathcal{E} \otimes \mathcal{F})_+ \rightarrow \mathcal{E}_+ $$ by: $$ \left(\tau f \right)(x) = \int_F f(x, y) \mathrm{d}K(x) $$
It is clear that for any $f \in \left( \mathcal{E} \otimes \mathcal {F} \right)_+ $, regardless of s-finiteness of $K$, and for any $x \in E$ the number $ \left( \tau f \right)(x) $ is well definied (although it might be $ \infty $) - it's simply an integral of some positive function (specifically, the $x$-section of $f$) by some measure ($K(x)$).
The problem is to show that the function $ \tau f $ is $ \mathcal{E} $ - measureable.
The book's strategy is basically showing that the set: $$ \left \{ f \in \left( \mathcal{E} \otimes \mathcal {F} \right)_+ | \tau f \in \mathcal{E}_+ \right \} $$ is a monotone class, and then apply MCT.
the interesting step is showing that if $f$ and $g$ are bounded functions in the set such that $f \geq g$ then $f-g$ is in the set, and this is proven by calculation of $$ \tau \left( f - g \right) $$ We want to say $$ \tau \left( f - g \right) = \tau f - \tau g $$ but this might be wrong in general, because it's possible that $\tau f(x) = \tau g(x) = \infty $ for some point $x \in E$ and the right hand side is not defined.
If we however require that the kernel is finite, then it cannot happen and the proof proceeds smoothly.
Even though I think I understand why the hypothesis that $K$ is s-finite is crucial to the claim that $\tau f$ is $\mathcal{E}$ - measureable, I fail to find a fitting counter-example.
TL;DR
Can you help me find a non-s-finite transition kernel $K$ from $(E, \mathcal{E})$ to $(F, \mathcal{F})$ and a postive measureble function: $$ f: (E \times F, \mathcal{E} \otimes \mathcal{F}) \rightarrow [0, \infty] $$
such that the function: $$ \tau f: E \rightarrow [0, \infty] $$
defined by: $$ \tau f(x) = \int_F f(x, y) \mathrm{d} K(x) $$
is not $\mathcal{E}$-measureable?