Proof of measurability in a proof

Question

I am trying to understand a proof in a book (A. Weir, General Integration and Measure, p.111). There $\mu$-measurability of a function $f$ can be proved by checking if $\text{mid}(-g,f,g)$ is in $L^1$ for all positive $g∈L^1$ (p.110). Then it is described that you can replace the functions $g$ by $K\chi_A$ with $\chi_A∈ L^1$. Now I don't understand, why you can conclude if $\min(\chi_E,\chi_A)$ is in $L^1$ for all $\chi_A∈ L^1$, then $\chi_E$ is in $L^1$ as on page 111. If you use the definition you can show that $\text{mid}(-K\chi_A,\chi_E,K\chi_A) = \min(\chi_E,K\chi_A)$. But how do I now that the functions $\min(\chi_E,K\chi_A)$ are integrable when I only know that $\min(\chi_E,\chi_A)$ are integrable? Has anyone an idea? Thank you!