摘要：Kronecker’s lemma gives a condition for convergence of partial sums of realnumbers, and for example can be used in the proof of Kolmogorov’s strong la... 阅读全文

摘要：Given $0<s<u<v<1$ and $s<t<v$, one can construct a Cantor set $E\subset [0,1]$ such that $\dim_H E=s, \dim_P E=t, \underline{\dim}_B E=u$ and $\overli... 阅读全文

摘要：Let $X=\{0, 1,1/2,\cdots, 1/n\cdots\}.$ Then, we have $\dim_X E=\dim_P X=0, \dim_B X=1/2$ and $\dim_A X=1.$Now we will prove that $\dim_AX=1.$Recall t... 阅读全文

摘要：The second Borel-Cantelli lemma:Let $(\Omega, \mathcal{F}, P)$ is a probability space. If the events $A_n$ are independent then $\sum P(a_n)=\infty$ i... 阅读全文

摘要：1. There exists compacet set $E\subset [0,1]$ with $\dim_H E<\dim_P E$ (in fact, for any $\beta\in(0,1)$, $\dim_HE=0, \dim_P E_{\beta}=\beta$ ), and u... 阅读全文

摘要：Let $X$ be a totally bounded metric space.(1) If $X$ is compact and if $\overline{\dim}_MU\ge s$ for every non-empty open set $U\subset X,$ then $\dim... 阅读全文