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:

\(\ds \map \mu F\)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \map \mu {E_n}\)

Measure of Limit of Decreasing Sequence of Measurable Sets

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \map \mu {F_{n + m} }\)

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \map \mu {F_n}\)

Tail of Convergent Sequence

$\blacksquare$