Activity for Ooker
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #296071 |
> set of all _actual_ bodies
Why do you emphasize "actual"? Did you simply mean whole, unseparated body, where body parts are not defined?
> set of all bodies in some probability distribution
Can you give an example of this? (more) |
— | 4 months ago |
| Comment | Post #296071 |
Well, I was thinking about splitting a human body into different parts: head, torso, limbs, genitals. I mistook body with torso in the previous comment. But I realize there are different σ-algebras that we can talk at the same time, and this can make confusion. For example, the body can be the wh... (more) |
— | 4 months ago |
| Comment | Post #296071 |
> If they were in the codomain of the measure then what would be in the domain?
Well, the domain of the measure is a σ-algebra over a set? So I understand that it makes no sense to say $σ(\text{height} < 1.72 \text{ m})$, but it makes sense to say $σ_\text{height}(\text{body}) = 1.72 \text{ m... (more) |
— | 4 months ago |
| Comment | Post #296071 |
I'm not sure why closing the question and asking a new one is better, as we can edit it and there is no answer yet? Anyhow, I think I should make sure my understanding is correct before asking a new one, so that it contain less unclear terms and misunderstandings.
This idea came from two thin... (more) |
— | 4 months ago |
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| Comment | Post #296071 |
In my understanding, a measure is a function from a σ-algebra (its domain) to $\mathbb R$ (its codomain). The event of getting what side of the dice is in the domain; the number showing up in that side is in the codomain. Is that correct?
I didn't say we to equip a superspace $X$ with properti... (more) |
— | 4 months ago |
| Comment | Post #296071 |
I see. Are there any X, Y, Z, T that are superspaces of $\mathbb R$ that we can use here?
As for the space of harmonic functions, I don't mean it should become $\mathbb R$. Instead, as it is countably additive, then I guess we can use it as an alternative codomain instead of $\mathbb R$. Which... (more) |
— | 5 months ago |
| Comment | Post #296071 |
### "Randomly pick one"
In the previous question you say that:
> the expectation is simply an integral (or, rather, "simply"...), you can certainly generalize it to any circumstance where integration makes sense
So I understand that we have the freedom to pick any integral that makes sense ... (more) |
— | 5 months ago |
| Comment | Post #296071 |
### "Become $\mathbb R$"
We know that $\mathbb R$ is a Hilbert space with many properties that other Hilbert spaces doesn't have. Then:
- If we remove those properties (de-equip them), then we have a superspace of $\mathbb R$, which still is a Hilbert space, which we will call as X
- If we re... (more) |
— | 5 months ago |
| Comment | Post #296071 |
### "Equip more properties"
Say you pick a metric space as a codomain of the measure. If you define a norm and state that it's complete, then you have a Banach space. That's "equip the norm and complete properties" to the metric space. If you define an inner product, then you have a Hilbert spac... (more) |
— | 5 months ago |
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| Comment | Post #296070 |
well at first I think it should be quick, but perhaps having a separated question is best indeed: [How important results in classical statistics be generalized when codomain of the measure is not $\mathbb R$?](https://math.codidact.com/posts/296071) (more) |
— | 5 months ago |
| Edit | Post #296071 | Initial revision | — | 5 months ago |
| Question | — |
How important results in classical statistics be generalized when codomain of the measure is not $\mathbb R$? From Can expectation be calculated if the events aren't numbered?, I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it ... (more) |
— | 5 months ago |
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| Comment | Post #296070 |
Can you give an example on how the expectation is constructed in a Banach space, maybe the $l^p$ spaces? Can you do it for other generalizations like variance, PDF, CDF, Gauss distribution, central limit theorem, etc. too? (more) |
— | 5 months ago |
| Edit | Post #296069 | Initial revision | — | 5 months ago |
| Question | — |
Can expectation be calculated if the events aren't numbered? Assuming I have a dice whose sides are named, not numbered. Can I have some way to describe the expectation of it? Or must I numbered the events in order to get the mean? To put it in another way: why must the codomain of the measure) be $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), ... (more) |
— | 5 months ago |