On the wikipedia page Measure(mathematics), it only gives the following inequality for the measure of infinite unions of finite sets:
$$ \mu( \bigcup_{i=0}^\infty E_i) \le \sum_{i=0}^\infty \mu( E_i) $$
But really I wasn't looking for that and I'm not well versed in measure theory, so just assume my logic is based on the discrete measure where each set is a set of integers and $\mu$ is the number of elements in the set. I don't need an infinite case, only one exact equation that is extensible for $n$ many sets, such that if I needed the infinite case, I could take the limit.
So if my question feels vague, I certainly apologize and I am exceptionally grateful to each of my readers.
Measuring a simple union offers the following easily recognizable result. $$ \mu(A \cup B) = \mu(A) - \mu(A \cap B) + \mu(B) $$ Letting $B = C \cup D$, $$ \mu(A \cup B) = \mu(A \cup C \cup D) = \mu(A) - \mu(A \cap (C \cup D)) + \mu(C \cup D)$$ and $A \cap (C \cup D) = (A \cap C) \cup (A \cap D) \implies$ $$\mu(A) - \mu(A \cap (C \cup D)) + \mu(C \cup D) = \mu(A) - \mu((A \cap C) \cup (A \cap D)) + \mu(C \cup D) $$ $$ = \mu(A) - \mu(A \cap C) + \mu((A \cap C)\cap(C\cap D)) - \mu(A \cap D) + \mu(C \cup D) $$ Observing $A \cap C \cap D = (A\cap C)\cap (C\cap D)$ and $\mu(C \cup D) = \mu(C) - \mu(C \cap D) + \mu(D)$
We find the following expression for the exact measure of a three term union:
$$ \mu(A \cup C \cup D ) = $$ $$ \mu(A) + \mu(C) + \mu(D) - \mu(A \cap C) - \mu(A \cap D) - \mu(C \cap D) + \mu(A \cap C \cap D) $$
I can't bear to bore you by continuing on to the four term union. But we can clearly see that whether we add or substract a term is directly related to how many sets are under intersection in the measure: addition for odd and subtraction for even, and each intersection is taken from each subset of sets in the covering giving this for the final equation -- the exact measure of the covering
-- albeit requiring foreknowledge of the measures of the intersections.
$$ \mu( \bigcup_{t\in T} E_t) = \sum_{i=0}^{\mu(T)}(-1)^{i+1}[\sum_{\mu(F) = i, \forall F \subseteq T} \mu( \bigcap_{t\in F} E_t )] $$
I should ask: can this be found in a textbook somewhere that I could verify my result?