$X$ independent of $Y$ conditional on $Y$ in some subset of the domain?

Let $$X,Y,\epsilon:\Omega\to \mathbb R$$ be random variables. Let's say that $$X=\text{sign} (Y) +\epsilon$$. Then $$X$$ is not independent of $$Y$$. However, we have all the information about $$Y$$ that we need for $$X$$ if we just know whether $$Y\in \mathbb R_{< 0}$$ or $$Y\in \mathbb R_{\geq 0}$$. That is, once we know that $$Y$$ is in one of those subsets, then $$X$$ is independent of further information about $$Y$$.

Is there a generally accepted name for this type of "partial" independence?

• How about conditional independence? – whuber Oct 11 at 14:39
• @whuber, I may be misinterpreting you, but is "How about" supposed to mean "you should have known that it is conditional independence"? In any case, conditional independence is more general, since $X$ and $Y$ can be conditionally independent conditional on an unrelated third variable $Z$, rather than conditional on $Y$ being in a subset of its domain. – user600667 Oct 11 at 15:47
• Define the third variable as the indicator of the conditioning event. I am trying to understand whether, and if so how, your question differs from inquiring about conditional independence. From your comment it sounds like there is no difference. – whuber Oct 11 at 16:15
• @whuber, like I said, conditional independence is more general. I'm wondering if there is a name for this special case......... – user600667 Oct 11 at 16:51