why double Sigma summation? The lecture slides is about covariance. One page it is


on the other page, it is



Why there are double Sigma summation on the first one?Is it just a typo?
I think a very clear definition of covariance is:

$Cov(X,Y) = \sum\limits_{x \in X}\sum\limits_{y \in Y} (x- \mu_X)(y-\mu_Y)f(x,y)$

If you define $S = X \times Y = \{(x,y) | x \in X, y \in Y\}$ (cartesian product).

Then you can rewrite the definition as:

$Cov(X,Y) = \sum\limits_{(x,y) \in S} (x- \mu_X)(y-\mu_Y)f(x,y)$

I think the first equation in your post is a confusion between the two definitions written in this answer.
It is not even clear to me what summation sign $(x,y) \in S$ belongs to.
However, a lot of notation abuse like this is made, if it is clear what is being meant. Which was obviously not the case here since you were confused.
It is a double summation, one over $x$ and one over $y$, so strictly speaking you need two $\Sigma$ signs on both slides. 

Sometimes people leave only one just for easier notation.