Measures & Measure Spaces

A σ-algebra tells us which sets are measurable. A measure tells us how big each one is. Given a sample space \Omega and a σ-algebra \mathcal{F}, a measure is a function

\mu : \mathcal{F} \to [0, \infty]

assigning a non-negative size to each measurable set, subject to two requirements: the empty set has zero size, \mu(\emptyset) = 0, and countable additivity (σ-additivity) — for any pairwise disjoint sets A_1, A_2, \dots \in \mathcal{F},

\mu\!\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu(A_i).

The triple (\Omega, \mathcal{F}, \mu) is called a measure space.

Two properties that follow

Additivity has immediate consequences. If A \subseteq B then B = A \cup (B \setminus A) is a disjoint split, so \mu(B) = \mu(A) + \mu(B\setminus A) \ge \mu(A). This is monotonicity:

A \subseteq B \;\Longrightarrow\; \mu(A) \le \mu(B).

And even when sets overlap, throwing away the double-counting can only shrink the total, so we still have countable subadditivity: \mu\!\left(\bigcup_i A_i\right) \le \sum_i \mu(A_i).

Additivity, made visible

The cleanest measure is length on the real line. Lay down two intervals that do not overlap; their lengths simply add. Step through to see \mu(A) = 2 and \mu(B) = 3 combine into \mu(A \cup B) = 5.

The measures you will meet

Lebesgue measure on \mathbb{R} — the generalisation of length (and of area, volume in higher dimensions). It is the measure that underpins integration, giving \mu\big([a,b]\big) = b - a.
Counting measure — \mu(A) = |A|, the number of elements of A (and \infty if A is infinite). On a finite set this is the natural way to "size" an event by how many outcomes it holds.

A measure with the extra normalisation \mu(\Omega) = 1 is exactly a probability measure — the subject of the next concept.