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.