Real analysis : pages from year three of a mathematical blog

ly characterized as Stone spaces such that the closure of a countable union of clopen sets is clopen. A (concrete) measure space (X,B, μ) is a concrete σ-algebra (X,B) together with a countably additive measure μ : B → [0,+∞]. One can similarly define an abstract measure space (B, μ) (or measure algebra) to be an abstract σ-algebra B with a countably additive measure μ : B → [0,+∞]. (Note that one does not need the concrete space X in order to define the notion of a countably additive measure.) One can obtain an abstract measure space from a concrete one by deleting X and then quotienting out by some σ-ideal of null sets—sets of measure zero with respect to μ. (For instance, one could quotient out the space of all null sets, which is automatically a σ-ideal.) Thanks to the Loomis-Sikorski representation theorem, we have a converse: Exercise 2.3.3. Show that every abstract measure space is isomorphic to a concrete measure space after quotieting out by a σ-ideal of null sets (where the notion of morphism, isomorphism, etc. on abstract measure spaces is defined in the obvious manner.) Notes. This lecture first appeared at terrytao.wordpress.com/2009/01/12. Thanks to Eric for Remark 2.3.11, and for the functoriality remark in Remark 2.3.13. Eric and Tom Leinster pointed out a subtlety that two concrete Boolean algebras which are abstractly isomorphic need not be concretely isomorphic. In particular, the modifier “abstract” is essential in the statement that “up to (abstract) isomorphism, there is no difference between a concrete Boolean algebra and an abstract one.” Author's preliminary version made available with permission of the publisher, the American Mathematical Society