Mass Transference Principle: From balls to arbitrary shapes: Measure theory

Abstract

Following a work of Koivusalo and Rams, by a further generalization of the singular value function, we extend the Mass Transference Principle set up by Beresnevich and Velani to lim sup sets generated by open sets of arbitrary shapes. For a Borel set E⊂Rd with d≥1 and a dimension function f, define a generalized singular value function asφf(E):=supμ∈M(E)⁡infx∈E⁡infr>0⁡f(r)μ(E∩B(x,r)), where M(E) represents the collection of all Borel probability measures supported on E and B(x,r) denotes the ball of center x and radius r. Let {Bi}i∈N be a sequence of balls in [0,1]d with radius rBi→0 as i→∞. Assume that λ(lim supi→∞Bi)=1, where λ is the Lebesgue measure in Rd. Let {Ei}i∈N be a sequence of open sets such that Ei⊂Bi and limi→∞⁡λ(Bi)λ(Ei)=∞. Let f be a dimension function such that f(r)/rd is monotonically decreasing and limr→0⁡f(r)rd=∞ and φf(Ei)≥λ(Bi) for each pair (Ei,Bi). Then, for any ball B in [0,1]dHf(B∩lim supi→∞Ei)=Hf(B).