Question involving integration of nonnegative measurable functions using simple functions

Question

Question: I am working on a problem in Folland's Real Analysis (specifically, 2.16) and I want to justify something: Suppose $f$ is a measurable nonnegative function and $\int f<\infty$. Let $\epsilon >0$. Then, there exists simple function $\phi=\sum_n a_n\chi_{E_n}$ such that $0\leq \phi\leq f$. So, by monotonicity, I can say $\int\phi \leq \int f$. It would be terrific if I could then say that for all $\epsilon>0$, we have $(\int f)-\epsilon<\int\phi$ (note that all integrals so far are taken over our general measurable set $X$). But, can I say this? What if $\epsilon$ is REALLY small.. then how can I essentially "flip" the integral $\int\phi\leq \int f$ when subtracting $\epsilon$ from the RHS?