... and taking intersections This shows M is a σ-algebra, and so M ⊂ A Suppose (X, A, µ) and (Y, B, ν) are two measure spaces, i.e., A and B are σ-algebras on X and Y , resp., and µ and ν are measures ... nonnegative, f≥ f≥ An µ(An ), n a contradiction Product measures If A1 ⊂ A2 ⊂ · · · and A = ∪∞ Ai , we write Ai ↑ A If A1 ⊃ A2 ⊃ · · · and i=1 A = ∩∞ Ai , we write Ai ↓ A i=1 Definition M is a monotone ... whenever the Ai are disjoint and all the Ai are in A ∞ i=1 µ(Ai ) Definition Let µ be a signed measure A set A ∈ A is called a positive set for µ if µ(B) ≥ whenever B ⊂ A and A ∈ A We define a negative...