Just heard a nice talk by Simon Henry about measure theory set up in Boolean topos theory (his main result is to identify Tomita-Takesaki-Connes’ canonical outer automorphisms on $W^\ast$-algebras in the topos language really nicely…).

I have to rush to the dinner now. But to remind myself, I have added cross-links between *Boolean topos* and *measurable space* and for the moment pointed to

- Matthew Jackson,
*A sheaf-theoretic approach to measure theory*, 2006 (pdf)

for more. Simon Henry’s thesis will be out soon.

Have to rush now…

]]>In order to un-gray a link at *conditional expectation* I created a minimum for *pushforward measure*.

I started a bare minimum at *quantum probability* (redirecting *noncommutative probability space* etc.)

Some entries have long been secretly referencing such an entry, and I have cross-linked accordingly, for instance from *von Neumann algebra* and *quantum computing*.

I had the feeling somewhere we already had a detailed account of probability theory dually in terms of von NNeumann algebras, but if we do I didn’t find it(?)

]]>Hello. I would like to create a page about valuations, the concept in measure theory, which is distinct from valuations in ring theory. Is there an nlab-equivalent of Wikipedia’s disambiguation pages? Or what is the practice here, in case two concepts have the same name?

Thanks.

]]>I wanted to be able to point to *[[expectation value]]* without the link being broken. So I added a sentence there, but nothing more for the moment.

I have created an entry *transgression of differential forms* that discusses the concept using the topos of smooth sets. Apart from the traditional definition as $\tau_{\Sigma} \coloneqq \int_\Sigma ev^\ast$ the entry considers the formulation as

which simply forms the internal hom into the classifying map $X \to \mathbf{\Omega}^n$ of a differential form. I have spelled out the proof that the two definitions are equivalent.

Then the entry contains statement and proof of the situation of “relative” transgression over manifolds with boundary. (This is what yields, when applied to Lepage forms, Lagrangian correspondences between the phase spaces with respect to different Cauchy surfaces, which is what I currently need this material for in the exposition at *A first idea of quantum fields*.)

Finally there are two examples, a simplistic one and an simple but interesting one related to Chern-Simons theory. These two examples I had kept for a long time already at *geometry of physics – integration* in the section “Transgression”. That section I have now expanded accordingly, its content now coincides with the entry transgression of differential forms.

I have split-off *Feynman diagram* from *perturbation theory*, gave it a brief Idea-section and added a pointer to an insightful reference.

started a bare minimum at *Bloch region*

created *volume*, just for completeness

stub for *[[moment]]*, just for completeness