Measure theory MOC

Measure space

A measure space consists of a measurable space and a measure on that space. A measurable space consists of a set and a σ-algebra on that set . A measure on a measurable space is a function satisfying measure

1. non-negativity (unless Signed measure): for all .
2. empty set has zero measure¹:
3. σ-additivity: for any with . By induction the same holds for any countable collection of pairwise disjoint sets.

Thus a measure space generalises volume in the same way that a metric space generalises length.

Properties

From the above axioms it follows

1. monotonicity:
2. countable subadditivity: Let be a countable (or finite²) sequence of measurable sets, then

Proof of 1–2

Let be measurable sets such that . Then by ^M5 and , so by ^M3 , wherefore , proving ^P1.

Now let Let be a countable sequence of measurable sets. Then

and by ^P1

hence

applying this argument inductively proves ^P2.

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Footnotes

1. If at least one has finite measure, then this follows from σ-additivity since . ↩

2. Just give the sequence trailing . ↩