The delta's stand are point measures. \delta_x has mass 1 at the point x and 0 everywhere else.
Therefore \hat\mu is the measure that has mass 1/J at each x_j and 0 everywhere else.

These point measures are also called "dirac delta" and the notation using the delta is common in my field, but it is rather confusing to just use them like that without explanation, thanks a lot for pointing that out. Will update that!