A lemma on a finite union-closed family of finite sets and its applications

Suppose that $\mathcal{F}$ is a finite union-closed family of sets with ${\cup }_{A\in \mathcal{F}}A=\{1,2,\dots ,m\}$ and $m\ge 2$. Fix $i\in \{1,2,\dots ,m\}$ and denote $\mathcal{G}:=\{A\mathrm{\backslash }\{i\}:A\in \mathcal{F}\}$. For $j\in \{1,2,\dots ,m\}\mathrm{\backslash }\{i\}$, let ${\mathcal{G}}_j:=\{A\in \mathcal{G}:j\in A\}$ and ${\mathcal{F}}_j:=\{A\in \mathcal{F}:j\in A\}$. In this note, we will prove a lemma which says that if $\frac{\mathrm{\mid }{\mathcal{G}}_j\mathrm{\mid }}{\mathrm{\mid }\mathcal{G}\mathrm{\mid }}\ge c(c\in (0,1])$, then $\frac{\mathrm{\mid }{\mathcal{F}}_j\mathrm{\mid }}{\mathrm{\mid }\mathcal{F}\mathrm{\mid }}\ge \frac{1}{1+2(1-c)\mathrm{/}c}$. Several applications of this lemma will be given.