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A general principle for Hausdorff measure

We introduce a general principle for studying the Hausdorff measure of limsup sets. A consequence of this principle is the well-known Mass Transference Principle of Beresnevich and Velani (2006). https://arxiv.org/pdf/1808.02135.pdf

The main result is as follows.

\begin{theorem}[Hussain-Simmons, 2018]\label{HS:thm}
Fix , let be a sequence of open sets in an Ahlfors -regular metric space , and let be a dimension function such that

(1)

Fix , and suppose that the following hypothesis holds:
\begin{itemize}
\item[(*)] For every ball and for every , there exists a probability measure with , such that for every ball , we have

(2)

\end{itemize}
Then for every ball ,

In particular, if the hypothesis \text{(*)} holds for all , then

\end{theorem}

The condition \eqref{f2} is a natural condition which implies that . A consequence of this theorem is the celebrated Mass Transference Principle of Beresnevich-Velani (2006).