According to Definition 2.7 in Ergodic Theory: with a view towards Number Theory, the systems $(X, \mathcal{B}_X, \mu, T)$ and $(Y, \mathcal{B}_Y, \nu, S)$ are isomorphic when there is a $X' \in \mathcal{B}_X$ and a $Y' \in \mathcal{B}_Y$ with $\mu(X') = \nu(Y') = 1$, $TX' \subset X'$, $SY' \subset Y'$, and an invertible measure-preserving $\phi: X' \to Y'$ $$ \phi \circ T(x) = S \circ \phi(x) $$ for all $x \in X'$. The book does not define isomorphism of measure spaces. I assume that it is the same as Definition 2.7, with the identity in place of $S$ and $T$.
Now, Exercise 2.1.1 asks me to show that $(\mathbb{T}, \mathcal{B}_{\mathbb{T}}, m_{\mathbb{T}})$ is isomorphic to $(\mathbb{T}^2, \mathcal{B}_{\mathbb{T}^2}, m_{\mathbb{T}^2})$, where $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and the measures $m_{\mathbb{T}}$ and $m_{\mathbb{T}^2}$ are the Haar measures in $\mathbb{T}$ and $\mathbb{T}^2$ with their usual group structure.
How do I do that?
This is not "homework", but it could be...