Measures

Assigning a consistent size to sets, and everything the two axioms force.

The idea

A measure is the common generalization of length, area, volume and probability: a rule that assigns to each set a non-negative size, with the size of a whole equal to the sum of the sizes of its disjoint parts.

Fix a set $X$ and a $\sigma$-algebra $\mathcal{M}$ of subsets of $X$ — the sets to be measured. A measure is a function $\mu \colon \mathcal{M} \to [0, \infty]$ with $\mu(\varnothing) = 0$ that is countably additive: for every sequence $A_{1}, A_{2}, \dots$ of pairwise disjoint members of $\mathcal{M}$, $\mu\left(\bigcup_{n=1}^{\infty} A_{n}\right) = \sum_{n=1}^{\infty} \mu(A_{n}).$ The triple $(X, \mathcal{M}, \mu)$ is a measure space.

The value $\infty$ is allowed: the whole real line has infinite length, and the definition accommodates that rather than excluding it. The disjointness is required: for overlapping sets the equation is false, since the right side counts each overlap twice and the left side once.

Every further property a notion of size should have follows from these axioms. Additivity over finitely many disjoint sets, monotonicity — $A \subseteq B$ forces $\mu(A) \le \mu(B)$ — and continuity along an increasing sequence of sets are consequences, not extra assumptions.

Ways to work on it

• Walkthrough. The two axioms, the properties they force, and the standard examples.
• Proof. See why a measure respects increasing limits — the disjoint-shell picture.
• Practice. Compute a measure under Lebesgue measure, counting measure, or a point mass.
• Hardest. Cover the rationals with a small set, and find where continuity from above breaks.

Not sure where to start? Take the ten-question placement test.