Let ((X, mathscr{A}, mu)) be a measure space and (A_{1}, ldots, A_{N} in mathscr{A}) such that (muleft(A_{n}ight)

Question:

Let $(X, \mathscr{A}, \mu)$ be a measure space and $A_{1}, \ldots, A_{N} \in \mathscr{A}$ such that $\mu\left(A_{n}ight)<\infty$. Then

\[
\mu\left(\bigcup_{n=1}^{N} A_{n}ight) \geqslant \sum_{n=1}^{N} \mu\left(A_{n}ight)-\sum_{1 \leqslant n\]

[ show first an inequality for indicator functions.]

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