Exchange of expectations in derivation of bias-variance decomposition

[kynnjo]

[I originally posted this question to the wrong sub-forum. My apologies.]

In the derivation of the bias-variance decomposition (on p. 63), there is a step in which the taking of expectation wrt \mathcal{D} and wrt \mathbf{x} are exchanged.

It's not clear to me that these two expectations commute: the choice of \mathbf{x} depends on the choice of \mathcal{D}, and viceversa.

I would appreciate a clarification on this point.

Thanks in advance,

kj

P.S. Pardon the "raw LaTeX" above. Is there a better way to include mathematical notation in these posts.