Let X1 , X2 , . be a sequence of independent, identically distributed() random variables each taking values 0, 1, a with equal probability 1/3. Let µ be the ∞ −n probability measure induced by S = Xn . Let α(s) ((s), α(s)) denote n=1 3 the local dimension (resp. lower, upper local dimension) of s ∈ supp µ, and let α = sup{α(s) : s ∈ supp µ}; α = inf{α(s) : s ∈ supp µ}