Let $(X,\Sigma,\mu)$ be a measure space and $\phi$ and $\psi$ nonnegative simple function on $X$. If $\alpha$ and $\beta$ are positive real numbers, then
$$ \int_X[\alpha \psi +\beta \phi]d\mu=\alpha \int_X \psi d\mu +\beta \int_X \phi d\mu $$.
In showing the property linearity above, we need to consider the case where $\psi$ or $\phi$ is positive on a set of infinite measure. In the proof given by Royden and Fitzpatrick in Real Analysis book,it says if either $\psi$ or $\phi$ is positive on a set of infinite measure, then the linear combination $\alpha \psi+\beta \phi$ has the same property and therefore each side of the equation is infinite.Can someone show a thorough proof of this part?