Say that $E$ is of finite measure and we have closed subsets $B_1, B_2$ such that $|E - B_1| < \epsilon_1$, $|E - B_2| < \epsilon_2.$ How can I write $|E - B_1 \cup B_2|$ in relation to $\epsilon_1, \epsilon_2?$
2026-04-26 06:11:56.1777183916
Computing the measure of a set difference
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We have $E - B_1 \cup B_2 \subset E - B_1$ and $E - B_1 \cup B_2 \subset E - B_2$, so by monotonicity, $$ |E-B_1 \cup B_2| \le |E - B_1| \\ |E-B_2 \cup B_2| \le |E-B_2| $$