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 A1,A2,...∈B are disjoint sets and A is their union, then μ(A) = ∑n≥1 μ(An).

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.