# Measure of Limit of Decreasing Sequence of Measurable Sets

## Theorem

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

Let $E \in \Sigma$.

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

$E_n \downarrow E$

where $E_n \downarrow E$ denotes the limit of decreasing sequence of sets.

Suppose also that $\map \mu {E_1} < \infty$.

Then:

$\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$

### Corollary

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

$\sequence {E_1 \setminus E_n}_{n \mathop \in \N}$ is increasing.

Further, we have:

 $\ds \bigcup_{n \mathop = 1}^\infty \paren {E_1 \setminus E_n}$ $=$ $\ds E_1 \setminus \paren {\bigcap_{n \mathop = 1}^\infty E_n}$ De Morgan's Law for Set Differences: General Case $\ds$ $=$ $\ds E_1 \setminus E$ Definition of Limit of Decreasing Sequence of Sets

so:

$E_1 \setminus E_n \uparrow E_1 \setminus E$

So, from Measure of Limit of Increasing Sequence of Measurable Sets, we have:

$\ds \map \mu {E_1 \setminus E} = \lim_{n \mathop \to \infty} \map \mu {E_1 \setminus E_n}$

From Measure of Set Difference with Subset, we have:

$\ds \map \mu {E_1} - \map \mu E = \lim_{n \mathop \to \infty} \paren {\map \mu {E_1} - \map \mu {E_n} }$

since $\map \mu {E_1} < \infty$.

From the Difference Rule for Real Sequences, we then have:

$\ds \map \mu {E_1} - \map \mu E = \map \mu {E_1} - \lim_{n \mathop \to \infty} \map \mu {E_n}$

giving:

$\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$

$\blacksquare$