In measurability, you want a sigma algebra (i.e., a measurable space) in your domain, but a topological one in your codomain. Otherwise, if you don't have a notion of convergence in your codomain, you cannot really define a useful integral.

2015年12月7日 · Definitions of Measurability: Outer-inner measure convergence vs. Caratheodory criterion Hot Network Questions When were videos first commonly downloaded on the Internet?

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2017年2月23日 · Finally, everything above was given with real-valued functions and a Borel $\sigma$-field on the codomain in mind. Measurability depends as much on the codomain's $\sigma$-field as it does on the domain's, but the intuition is the same: follow the measurable and non-measurable sets to understand measurable and non-measurable functions.

2021年2月3日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2017年4月26日 · All random variables are defined on the same measurable space $(S,\mathcal{A}, P)$. The symbol $\mathcal{B}_{\mathcal{K}}$ denotes the Borel $\sigma$-algebra for the set $\mathcal{K}$.

Measurability of almost everywhere continuous functions. 0. Ensuring continuity and differentiability of a ...

And measurability is enough there. And an analytic function is automatically measurable with respect to every complete measure. $\endgroup$ – Michael Greinecker

2022年2月13日 · Also, usually a stochastic process does not assume product measurability, only adapted to certain filtration. The product measurability (also called progressively measurability) comes from additional path regularity condition, like càd or càg. $\endgroup$ –

2024年12月25日 · On the Lebesgue-measurability of the projective subsets of $\mathbb{R}^{n}$, the 4th page (pg. 570) of this article by Woodin says: Theorem (Godel). Assume ZFC is consistent. Then so is ZFC + “There is a nonmeasurable projective set”. So far so good, I think I understand this. What I don’t get is on the next page: