Suppose that $\mu$ is the measure on $(\mathbb{Z}^+,2^{\mathbb{Z}^+})$ defined by $$\mu(E)=\sum_{n\in\,E}\frac{1}{2^n}$$
Prove that for every $\epsilon>0$, there exists a set $E\subset\mathbb{Z}^+$ with $\mu(\mathbb{Z}^+\setminus\,E)<\epsilon$ such that $f_1,f_2,\dots$ converges uniformly on $E$ for every sequence of functions $f_1,f_2,\dots$ from $\mathbb{Z}^+$ to $\mathbb{R}$ that converges pointwise on $\mathbb{Z}^+$.