Abstract
This chapter surveys some of the interactions between measure theory and general topology. Fremlin's grand treatise on measure theory is used extensively as a reference. The two primary objects of study in measure theory are measure spaces and measure algebras. There are natural correspondences between measure spaces and measure algebras. Much of the theory of one object can be translated to the other, but not all of the results correspond and attention cannot be simply restricted to only one of them without losing some valuable aspect of the measure theory. Given a measure space (X, ɛ, μ), if ℕμ denotes the ideal of null sets (or negligible subsets) of X, that is, ℕ is null if there is an E ∈ɛ such that E ⊇N and μ (E) is 0. It is almost a first principle of measure theory that negligible sets can be ignored. Thus, in most cases (but not always), two sets that differ by a null set can be treated as the same set, and the measure algebra of a measure space (X, ɛ, μ) can be evaluated in this manner.
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