Coarsest $\sigma$-field of $\mathbb E(X\mid Y)$

Question

I was reading up on Conditional expectation and Sigma field and saw this statement which I dun really understand.

The coarsest $\sigma$-field for which $\mathbb E(X\mid Y)$ is a random variable is the $\sigma$-field generated by the $Y$ that is denoted by by $\sigma(Y)$

I don't really understand this statement. Need some explanation.

Answer 1

$\mathbb E(X\mid Y)$ is in particular, by definition, a real valued random variable which is $\sigma(Y)$-measurable.

Without further restrictions, we cannot assert that it is measurable with respect to a strictly smaller $\sigma$-algebra (for example take $Y=X$).