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 #m/def/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 σ-additivity , wherefore , proving monotonicity.

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

and by monotonicity

hence

applying this argument inductively proves countable subadditivity.

#state/tidy | #lang/en | #SemBr

Footnotes

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

  2. Just give the sequence trailing . ↩