How to construct an ergodic measure on canonical space

Question

Let $\mathbb{R}^{\mathbb{N}}$ be the canonical space, $\{x_{i}\}_{i\geq 1}$ be the canonical process. Define the shift operator: $\theta(x_{1},x_{2},...)=(x_{2},x_{3},...)$ and $\mathcal{G}$ be the invariant algebra with respect to $\theta$. My question is about how to construct an ergodic measure with respect to $\theta.$ In fact, given a probability $\mu$ on $\mathcal{B}(\mathbb{R})$, let $\mu^{\times\mathbb{N}}$ be the product measure of $\mu$ on $\mathbb{R}^{\mathbb{N}},$ then $\mu^{\times\mathbb{N}}$ is obviously an ergodic measure with respect to $\theta.$ This is because $\{x_{i}\}_{i\geq 1}$ is independent under $\mu^{\times\mathbb{N}}$, by zero-one law, $\mu(\tau)=\{0,1\},$ where $\tau$ denote the tail $\sigma$ algebra, note $\mathcal{G}\subseteq\tau$, thus $\mu({\mathcal{G}})=\{0,1\}.$ But can we construct an ergodic measure but NOT independent measure? Precisely, can we give a characterization of ergodic measures on $\mathbb{R}^{\mathbb{N}}$ with respect to $\theta$? Thanks!