Inclusion-exclusion for a nonempty finite family of finite-measure sets

Let $P(r)$ be the displayed inclusion-exclusion formula for a list of $r+1$ finite-measure measurable sets. It suffices to prove $P(r)$ for every $r\in \mathbb{N}$.

For $r=0$, both sides of $P(0)$ are $\mu (A_0)$, since the only nonempty subset of $\{0\}$ is $\{0\}$ and $(-1{)}^2=1$.

Fix $r$ and assume $P(r)$ for every list of $r+1$ such sets.

Put $U={\bigcup }_{i<r+1}{A}_i$. By [L2], $\mu (U)<+\mathrm{\infty }$, and [L1] applied to $U$ and $A_{r+1}$ gives $\mu (U\cup A_{r+1})=\mu (U)+\mu (A_{r+1})-\mu (U\cap A_{r+1})$ in $\mathbb{R}$.

The induction hypothesis expands $\mu (U)$ over the nonempty subsets of $\{0,\dots ,r\}$ and expands $\mu (U\cap A_{r+1})=\mu ({\bigcup }_{i<r+1}(A_i\cap A_{r+1}))$ over the same subsets; all intersections remain finite-measure.

In step 2.1, the terms from $\mu (U)$ are indexed by nonempty subsets not containing $r+1$, the term $\mu (A_{r+1})$ is indexed by $\{r+1\}$, and the negated terms from step 2.2 are indexed in the stated recursive order by the sets $J\cup \{r+1\}$ and acquire the sign $(-1{)}^{\mathrm{\mid }J\mathrm{\mid }+2}=(-1{)}^{\mathrm{\mid }J\cup \{r+1\}\mathrm{\mid }+1}$. Finite-sum splitting therefore gives $P(r+1)$.

By induction, $P(r)$ holds for every $r$, hence the stated formula holds for every nonempty finite family; the one-set boundary is step 1.2, and finiteness was used exactly in step 2.1 to permit subtraction.