A

measure space on a

set X is a

tuple (X,

**B**,μ) of the set X, a

σ algebra **B** on X, and a

nonnegative function μ:

**B**→[0,∞]

(yes, we allow "infinity", but if that bothers you pretend we don't). We require that μ be

σ additive (for

disjoint sets), i.e. that if A

_{1},A

_{2},...∈

**B** are

*disjoint* sets and A is their

union, then μ(A) = ∑

_{n≥1} μ(A

_{n}).

A measure attempts to capture our intuition of "quantity" -- length, area, volume and probability can all be defined as measures.

Due to technical difficulties, **B** will generally *not* contain all subsets of X. These difficulties are known to be unavoidable, at least if you accept the Axiom of Choice.