This should be a famous question on a (finite) algebra.
Let $\mathcal{F}$ be a collection of subsets of $\mathbb{N}$ defined by \begin{align*} \mathcal{F}=\left\{A \subset \mathbb{N} : \lim_{n \to \infty}\frac{\#(A\cap \{1,2,\ldots,n\})}{n} \text{ exists}\right\}. \end{align*} Here, we denote by $\#$ the counting measure.
Why $\mathcal{F}$ is a (finite) algebra on $\mathbb{N}$?
That is, I would like to know how to prove $A \cup B \in \mathcal{F}$ provided that $A,B \in \mathcal{F}$.
This should be equivalent to show that the cesaro mean $\{(1/n)\sum_{k=1}^n\textbf{1}_{A \cap B}(k)\}_{n=1}^\infty$ converges if the cesaro means $\{(1/n)\sum_{k=1}^n\textbf{1}_{A}(k)\}_{n=1}^\infty$ and $\{(1/n)\sum_{k=1}^n\textbf{1}_{B}(k)\}_{n=1}^\infty$. However, I don't feel that this claim holds true.
