Given some expression, for example a quadratic form $\langle X, AY \rangle$, where both $X$ and $Y$ are $N(0,I_n)$ random vectors. What do we mean by $\mathbb{E}_X\left[\langle X, AY \rangle\right]$.
Do we mean to condition on $Y$ and average over $X$? How should one think about such an expression; is it conditional expectation or something completely different.
Also how does it relate to Fubini's, for example
$\mathbb{E}_Y\mathbb{E}_X\left[\langle X, AY \rangle\right]$
Is this actually changing the order of two conditional expectations?
This question primarily stems from notation used in High-Dimensional Probability by Roman Vershynin.