1. Show that α, as defined in the proof of Theorem, is a countably additive measure even if µ and ν are not σ-finite.
2. If (Xi , Si , µi ) are measure spaces for all i in some index set I , where the sets Xi are disjoint, the direct sum of these measure spaces is defined by taking X = /i Xi , letting S :={ A ⊂ X : A ∩ Xi ∈ Si for all i }, and µ( A) := },i µi ( A ∩ Xi ) for each A ∈ S. Show that (X, S, µ) is a measure space.