Finite-exposure union bound: averaging conditional bounds over good atoms

Let $(\Omega,\mathcal F,\mu)$ be a probability space, $A$ a finite set and $f:\Omega\to A$ a map whose fibers $\{f=a\}=\{\omega\in\Omega: f(\omega)=a\}$ are measurable for every $a\in A$ (an exposure with finitely many possible values). Let $E\subseteq\Omega$ be an arbitrary event, $G\subseteq A$ a set of good values, and $\beta,\gamma\in[0,\infty]$. Assume

1. the exposure is bad with small probability: $\mu(\{\omega: f(\omega)\notin G\})\le\beta$;
2. on every good fiber, $E$ has small relative mass: $\mu(E\cap\{f=a\})\le\gamma\,\mu(\{f=a\})$ for every $a\in G$.

Then

The point is that the conditional bounds on the good atoms are averaged against the law of $f$, so no factor $|A|$ appears even though $A$ may be very large. This is how a failure probability is bounded by first revealing a finite amount of randomness (for instance, everything that is read at one point), discarding the unfavourable revelations, and then using a uniform conditional estimate on the favourable ones.

Formalization Note $E$ is not assumed measurable: for a non-measurable set, $\mu(E)$ is the outer measure, and only monotonicity and subadditivity are used. The bounds $\beta,\gamma$ are extended nonnegative reals (ENNReal).