Let $A \subset \nat$, then $A$ has **positive upper density** if $ \limsup_{n \to \infty}\frac{A \cap [1, n]}{n} > 0 $ then $A$ admits an arithmetic progression of length $k \in \nat$ for all $k \in \nat$. > [!theorem] > > Let $(\Omega, \cf, \bp, T)$ be an [[Ergodic Measure System|ergodic system]], $E \in \cf$ with $\bp(E) > 0$, and $k \in \nat$, then there exists $n \in \nat$ such that > $ > \bp\paren{\bigcap_{j = 0}^{k - 1}T^{jn}E} > 0 > $ Let $A \subset \nat$, and let $x_{A}: \nat \to \bracs{0, 1}$ by $x_A(n) = 1$ if $n \in A$, and $x_A(n) = 0$ otherwise. Let $X$ be the orbit of $x_A$ under the shift map on $\nat$. Suppose that $A$ has positive upper density, then there exists $\seq{N_j}$ such that $ \frac{1}{N_j}\abs{A \cap \braks{0, N_j}} \to \limsup_{n \to \infty}\frac{|A \cap [0, n]|}{n} $ as $j \to \infty$. For each $j \in \nat$, define $\mu_j = \frac{1}{N_j}\sum_{k = 1}^{N_j}\one_{\braks{y_k \in A}}$, then there exists a subsequence $\seq{\mu_{j_k}}$ such that $\mu_{j_k} \to \mu$ weakly as $k \to \infty$. The limit $\mu$ is a translation-invariant measure on $X$, which gives $x_A$ a positive measure.