Category:Measure of Limit of Decreasing Sequence of Measurable Sets

This category contains pages concerning Measure of Limit of Decreasing Sequence of Measurable Sets:

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:

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