Editing Measure (mathematics) (section)

==Basic properties==
Let <math>\mu</math> be a measure.

===Monotonicity===
If <math>E_1</math> and <math>E_2</math> are measurable sets with <math>E_1 \subseteq E_2</math> then
<math display=block>\mu(E_1) \leq \mu(E_2).</math>

===Measure of countable unions and intersections===
====Countable subadditivity====
For any [[countable]] [[Sequence (mathematics)|sequence]] <math>E_1, E_2, E_3, \ldots</math> of (not necessarily disjoint) measurable sets <math>E_n</math> in <math>\Sigma:</math>
<math display=block>\mu\left( \bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu(E_i).</math>

====Continuity from below====
If <math>E_1, E_2, E_3, \ldots</math> are measurable sets that are increasing (meaning that <math>E_1 \subseteq E_2 \subseteq E_3 \subseteq \ldots</math>) then the [[Union (set theory)|union]] of the sets <math>E_n</math> is measurable and
<math display=block>\mu\left(\bigcup_{i=1}^\infty E_i\right) ~=~ \lim_{i\to\infty}  \mu(E_i) = \sup_{i \geq 1} \mu(E_i).</math>

====Continuity from above====
If <math>E_1, E_2, E_3, \ldots</math> are measurable sets that are decreasing (meaning that <math>E_1 \supseteq E_2 \supseteq E_3 \supseteq \ldots</math>) then the [[Intersection (set theory)|intersection]] of the sets <math>E_n</math> is measurable; furthermore, if at least one of the <math>E_n</math> has finite measure then
<math display=block>\mu\left(\bigcap_{i=1}^\infty E_i\right) = \lim_{i\to\infty} \mu(E_i) = \inf_{i \geq 1} \mu(E_i).</math>

This property is false without the assumption that at least one of the <math>E_n</math> has finite measure. For instance, for each <math>n \in \N,</math> let <math>E_n = [n, \infty) \subseteq \R,</math> which all have infinite Lebesgue measure, but the intersection is empty.