How is mean independence defined?

Question

Suppose $X$ and $Y$ are random variables. As I understood it, mean independence is defined as follows

$Y$ is mean-independent of $X$ iff $\Bbb E[Y|X] = \Bbb E[Y]$

But my professor in class gave the following definition in class yesterday

$Y$ is mean-independent of $X$ iff $\Bbb E[Y|X] = c$, where $c \in \Bbb R$

Is this the same thing?

Like suppose $\Bbb E[Y|X] = c$, where $c \ne \Bbb E[Y]$.

EDIT:

Actually, my last statement can't be true by the law of iterated expectations since

$$E_X[E[Y|X]] = E[Y]$$

And by that law, then if $E[Y|X] = c$, $E_X[E[Y|X]] = E[Y] = c$

Answer 1

This is indeed the same thing, and follows from the properties of conditional expectation:

$$E[Y|X]=c$$

implies

$$E[Y]=E[E[Y|X]]=E[c]=c.$$

Answer 2

The notation of your professor means that E[Y|X] is a constant value, which does not depend of X, so Y is mean-independent of X.