Countable additivity

For pairwise disjoint measurable sets

E_{n}

, countable additivity means

μ (⋃_{n = 1}^{\infty} E_{n}) = \sum_{n = 1}^{\infty} μ (E_{n}) .

Table of contents
- Continuity from below of a measure Countable additivity
- Continuity from above of a measure Countable additivity

Continuity from below of a measure

If

E_{n} ↑ E

, then countable additivity gives

μ (E_{n}) ↑ μ (E)

, even when the limiting measure is infinite.

Continuity from above of a measure

If

E_{n} ↓ E

and

μ (E_{1}) < \infty

, then

μ (E_{n}) ↓ μ (E)

. Apply continuity from below of a measure to the increasing differences

E_{1} ∖ E_{n}

.

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