What does $\sigma(Y)$-measurable mean in this context of conditional expectation?

Question

I'm reading about conditional expectation in lecture note:

Could you please explain what $\sigma(Y)$-measurable means in this context? I saw in other textbook that $\sigma(Y)$ denotes the sigma-algebra generated by the collection $Y$ of sets. But here $Y$ is a random variable.

Thank you so much!

Answer 1

It means that there exists a measurable function $h\colon\Omega\to\mathbb R$ such that $$\mathbb P\bigl(\mathbb E[X\mid Y]=h(Y)\bigr)=1.$$

Answer 2

From @Norse's link, I get

$\sigma$ -algebra generated by random variable or vector Suppose $(\Omega, \Sigma, \mathbb{P})$ is a probability space. If $Y: \Omega \rightarrow \mathbb{R}^{n}$ is measurable with respect to the Borel $\sigma$-algebra on $\mathbb{R}^{n}$ then $Y$ is called a random variable $(n=1)$ or random vector $(n>1)$. The $\sigma$-algebra generated by $Y$ is $$\sigma(Y)=\left\{Y^{-1}(A)\mid A \in \mathcal{B}\left(\mathbb{R}^{n}\right)\right\}$$

I post it as an answer to peacefully close this question.