Definition: A measure on a σ-field F is a function μ: F --> [0,∞] satisying μ(∅) = 0 and σ-additivity: if A = ∪j=1..inf Aj with Aj being some disjoint sets in F, then μ(A) = Σj=1..inf μ(Aj).
A measure assigns a number to a set, with the null set getting 0 and the union of two sets getting the sum of their measures individually.

Properties: Any measure μ is:

1. monotone: AB implies μ(A) ≤ μ(B).
If one set is inside another, its measure is smaller than the set it is inside of.
2. continuous from below: if A1 &sube A2 ⊆ . . . is an increasing sequence of measurable sets, then μ(∪j=1..inf Aj) = limj-->inf μ(Aj).
The measure of the union of an infinite number of nested sets is the same as the limit of the measure as you take larger and larger sets.
3. conditionally continuous from above: if B1B2 ⊇ ... is a decreasing sequence of measurable sets and if the measures μ(Bj) are finite, then μ(∩j=1..inf Bj) = limj-->inf μ(Bj)
4. subadditive: if BA1A2 ∪ ... &cup An, then μ(B) ≤ μ(A1) + ... + μ(An).
If a set is contained in the union of a bunch of other sets, its measure is less than the sum of the measures of the other sets.
5. σ-subadditive: if B ⊆ ∪j=1..inf Aj, then μ(B) ≤ Σj=1..inf μ(Aj).

More: Some examples of measures are the Lebesgue measure the Hausdorff measure, and plain old Euclidean distance (on the appropriate sets). Measure theory is good for dealing with weird and pathological mathematical constructs like the Cantor set and Dirichlet function.