space on a set
X is a tuple
,μ) of the set X, a σ algebra B
on X, and a nonnegative function
→[0,∞] (yes, we allow "infinity", but if that bothers you pretend we don't)
. We require that μ be σ additive
sets), i.e. that if A1
sets and A is their union
, then μ(A) = ∑n≥1
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.