# Assumptions needed for independence of out-of-sample loss

I am reading Cross Validation with Confidence by Jing Lei. In section two he introduces cross-validation and makes a law of large numbers argument that CV gives you a good estimate.

Essentially, for a hold-out data point $$(X, Y)$$, we consider the loss $$\ell(\hat f(X), Y)$$ conditional on our fit model $$\hat f$$. The population equivalent is $$\mathbb{E} \left[ \ell(\hat f(X), Y) \vert \hat f \right]$$, which is what we would like to estimate.

Now, if we have multiple held-out data points, say $$(X_{n+1}, Y_{n+1}), (X_{n+2}, Y_{n+2}), ...$$, and we get corresponding estimates of out-of-sample loss $$\ell(\hat f(X_{n+1}), Y_{n+1}), \ell(\hat f(X_{n+2}), Y_{n+2}), ...$$, we can make an LLN argument that their average converges to our estimand provided $$\ell(\hat f(X_{n+1}), Y_{n+1})$$ and $$\ell(\hat f(X_{n+2}), Y_{n+2})$$, etc, are independent.

I see how $$\epsilon \perp X$$ is enough for independence of out of sample losses, but not how $$\mathbb{E}( \epsilon | X) = 0$$ (the condition in the text) is sufficient for this independence.