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
From Relative Complement of Decreasing Sequence of Sets is Increasing, we have:
- $\sequence {E_1 \setminus E_n}_{n \mathop \in \N}$ 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:
- $\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$
$\blacksquare$