Let $U$ and $X$ be random variables. I found that in the literature the mean independence assumption is sometimes stated as $\mathrm{E}[U|X=x] = 0$ for all $x$ in the support of $X$, and sometimes stated as $\mathrm{E}[U|X]=0$ with probability one (under $P_X$).
Is there any difference between these two concepts? Is one weaker than the other, or are they equivalent (or nonnested)? Any textbook reference/theorems for illustrating their differences will also be appreciated (unfortunately, I cannot find any).
My thought: Suppose $X$ is continuous. Consider the situation where the statement does not hold true at only a single point in the support of $X$, while it still holds true with probability one, I tend to think the former is stronger than the latter. But this is just a special case I could think of.