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Wow, that’s a lot!
Is this to be taken ironically? :-) I know, that it just says that some triangle is commutative, but I had to learn these topics in some detail at that time.
However, since the conditional expectation is just some integral- this universal property may be related to (be some tiny instance of) what Urs expanded on in his recent post on integration (which I haven’t read, yet).
Looks interesting, I need to have a closer look.
But first some more boring editorial notes:
Right at the beginning the map $Y$ needs to be introduced before the pushforward of the meansure along $Y$ is mentioned.
I have touched the formatting at conditional expectation a bit: added floatic TOC, added hyperlinks (some of them are now requesting the creation of further new entries on probability theory). Also prettified the formatting of the first commuting diagram; see the source code for how to use
\mathrlap
and
\mathllap
to make arrow-labels stick out to the right and left such as not to displace the arrow that they are labelling.
the map Y needs to be introduced
I have added this. The links to probability theory perhaps will have rather example character since for the theorem that the conditional expectation exists I think (at the moment) it is sufficient to require the restricted measure $P|_\mathfrak{S}$ to be $\sigma$-finite and we do not need it to be a probability measure. In this case the article could be renamed to conditional integral - but this is (as theorem 2 states) just an integral kernel. So for foundations of integration it is probably more interesting if the theorem of Radon-Nikodym has some formulation in terms of a factorization property.
The statement of the theorem reminds me of the Kan extension.
Okay, thanks. I’ll try to look at it in detail, maybe this evening.
By the way, I made conditional probability redirect to this entry. But maybe this deserves some more discussion.
It would be good to have an Idea-section at the beginning of the entry: “It is traditional to define that… . We show that this can be understood as…”, something like this, to give the reader an orientation for what to expect.
I made conditional probability redirect to this entry. But maybe this deserves some more discussion.
Conditional probability (of an event) is indeed defined as the conditional expectation of the indicator function (of the event).
It would be good to have an Idea-section at the beginning of the entry: “It is traditional to define that… . We show that this can be understood as…”, something like this, to give the reader an orientation for what to expect.
Of course we can write some motivation as in elementary stochastic (”Given an event A we ask for the probability of some event B provided that A is the case.”) but it would be nice to put it in a rather conceptual context.
I added a brief section on conditional expectation phrased in quantum/noncommutative probability theory.
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