Measure of Limit of Decreasing Sequence of Measurable Sets/Corollary

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $F \in \Sigma$.

Let $\sequence {F_n}_{n \mathop \in \N}$ be an decreasing sequence of $\Sigma$-measurable sets such that:


 * $F_n \downarrow F$

where $F_n \downarrow F$ denotes the limit of decreasing sequence of sets.

Suppose also that $\map \mu {F_m} < \infty$ for some $m \in \N$.

Then:


 * $\ds \map \mu F = \lim_{n \mathop \to \infty} \map \mu {F_n}$

Proof
Define the sequence $\sequence {E_n}_{n \mathop \in \N}$ by:


 * $E_n = F_{m + n}$

Then from Tail of Decreasing Sequence of Sets is Decreasing:


 * $\sequence {E_n}_{\mathop \in \N}$ is an decreasing sequence of $\Sigma$-measurable sets.

From Limit of Tail of Decreasing Sequence of Sets, we have:


 * $E_n \downarrow F$

with:


 * $\map \mu {E_1} = \map \mu {F_{m + 1} }$

Since $\sequence {F_n}_{n \mathop \in \N}$ is decreasing, we have:


 * $F_{m + 1} \subseteq F_m$

So from Measure is Monotone, we have:


 * $\map \mu {F_{m + 1} } \le \map \mu {F_m} < \infty$

so:


 * $\map \mu {E_1} < \infty$

Then: