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
Fubini's theorem only applies in a product space $Z=X\times Y $ when $X$ and $Y$ are independent, that is, if $P_Z(A\times B)=P_X(A)\times P_Y (B)$.
In case of non-independence:
$$\int f(x,y)dP_Z=\int \int f(x,y)dP_XdP_Y$$
is no longer valid, but this other one:
$$\int f(x,y)dP_Z=\int dP_X\int f(...