Quasi-Assouad dimensions for random measures supported on $[0,1{]}^d$

We introduce a probability distribution on $\mathcal{P}([0,1{]}^d)$, the space of all Borel probability measures on $[0,1{]}^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero lower quasi-Assouad dimension (hence the upper and lower Assouad dimensions are almost surely infinite or zero). We also indicate how the results extend to other Assouad-like dimensions.