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Consider a locally compact abelian group G and some character χ: G→T. The functional Mχ: f∈L1(G)↦∫fχ∈C is multiplicative: Mχ(f*g)=Mχ(f)Mχ(g), where f*g is the convolution of f and g. At the first glan …

There is a complete classification of probability spaces up to a measure-preserving isomorphism. Specifically, consider a category whose objects are triples (X,Σ,μ), where X is a set, Σ is a σ-algebr …

Yes, there is such a category, namely the category of measurable spaces and morphisms of measurable spaces equipped with fiberwise measures (also known as operator valued weights). See this answer for …

From the categorical viewpoint there is no difference, because the category of measurable spaces is equivalent to the category of complete measurable spaces with the equivalence given by the completio …

A conceptual answer can be given in the framework of this answer. The functor that sends a measurable space X to the set of random variables on X, i.e., equivalence classes of (unbounded) real or com …

Any smooth manifold has a canonical σ-ideal of negligible subsets, and μ must vanish on these. Apart from that, the Lie derivative of μ with respect to any smooth vector field must exist. This is ho …

Take the category of measurable locales, equip it with its natural Grothendieck topology, and take the topos of sheaves of sets on the resulting site. (Apply standard disclaimers about universes, coac …

According to Proposition 416Q(b) in Fremlin's Measure Theory, finitely additive functionals A→[0,∞) are in a canonical bijective correspondence with finite Radon measures on the Stone space Spec(A) of …

Yes, it's called a pushforward! For more details, see this answer: Conditional Expectation for $\sigma$-finite measures

One can define a reasonable notion of conditional expectation for arbitrary localizable measurable spaces, not necessarily σ-finite. This is explained in great detail in the answer to Is there an intr …

Perhaps it makes sense to mention this example: The category of measurable spaces is equivalent to the category of hyperstonean topological spaces and hyperstonean maps between them. To construct a m …

There are at least three different answers that can be given to this question, and in all three interpretations the answer essentially states that all maps are “open”, for the appropriate analogue of …

The spectrum of $L^\infty(R)$ is the hyperstonean space associated with the measurable space R. More information can be found in Takesaki's Theory of Operator Algebras I, Chapter III, Section 1.

To clarify Chris Heunen's answer, let me point out that most notions of measure theory have analogs in the category of smooth manifolds. For example, the analog of a measure space (X,M,μ), where X is …

Without loss of generality we can assume that the support of the measure equals $X$ (i.e., the measure is faithful), because we can always pass to the subspace defined by the support of the measure. T …