The theorem and part of its proof are given below:
My question is:
In the second paragraph, are they taking the infimum of the inequality in the first paragraph and because the left-hand side is already a number (the infimum) then the right-hand side is the only that is affected by taking the infimum? could anyone help me in answering this question , please?
The inequality $$ \int_E(f+g)\le \int_E\psi_1+\int_E\psi_2 $$ is true for all simple functions $\psi_1$ and $\psi_2$ with $f\le \psi_1$ and $g\le \psi_2$. Taking infimum among all such $\psi_1$ on both sides and then the infimum among all the $\psi_2$ gives $$ \int_E(f+g)\le \inf_{f\le\psi_1,\psi_1\text{simple}}\int_E\psi_1 +\inf_{f\le\psi_2,\psi_2\text{simple}}\int_E\psi_2=\int_Ef+\int_Eg. $$