Good Afternoon,
I am trying to understand this equality :
$$ \mathbb E \left [ \int_{-1/2}^{1/2} \int_{-1/2}^{1/2} d Z^*(f) d Z(f') \right ] = \int_{-1/2}^{1/2} d S^{(I)} $$
where $*$ stands for the complex conjugate, and where the $dZ^*$ and $dZ$ are orthogonal processes (and $L_1$ obviously for convergence sake), in other words :
$$ \mathbb E \left [ dZ^*(f) dZ(f') \right ] = 1_{f = f'} dS^{(I)}(f) $$
my concern is that I have the impression the above equality is "kind of" the same as:
$$ \int_{-1/2}^{1/2} \int_{-1/2}^{1/2} 1_{x=y} dx dy ( = 0) = \int_{-1/2}^{1/2} dx (\neq 0) $$
Can someone explain me why is the first equality true, and if there is any bit of information missing please let me know.