I am studying Ergodic Theory for the first time, and am using the book "Ergodic Theory with a view towards Number Theory" by Einsiedler and Ward.
I got stuck at the very first exercise problem, numbered 2.1.1. which says: Show that the spaces $(\mathbb T,B_{\mathbb T},m_{\mathbb T})$ and $(\mathbb T^2, B_{\mathbb T^2},m_{\mathbb T^2})$ are isomorphic as measure spaces, where $\mathbb T=\mathbb R/\mathbb Z$ and $m_{\mathbb T}$ is the Lebesgue measure on $\mathbb T$.
The book does not define isomorphic measure spaces, but I think it means I hae to give a bijective map $f:(\mathbb T^2, B_{\mathbb T^2},m_{\mathbb T^2})\to (\mathbb T,B_{\mathbb T},m_{\mathbb T})$ so that $f$ and $f^{-1}$ are measurable. Notice, $B_{\mathbb T}$ is the Borel $\sigma-$algebra on $\mathbb T$.
I also am not sure I understand what $\mathbb T^2$ means. Does it mean $\mathbb T\times \mathbb T$? That is $\{(x+\mathbb Z,y+\mathbb Z):x,y\in[0,1)\}$?
So I have two real numbers, $x$ and $y$, and I want to map them bijectively to a new number in $[0,1)$. Can I do that?
Should this at all be my approach?
Update
I read from Bogachev that two measure spaces $(X,A,\mu)$ and $Y,B,\nu)$ are isomorphic iff there exists a bijective map $\phi:X\to Y$ so that $\phi$ and $\phi^{-1}$ are measurable and there exist $X'\subset X$, $Y'\subset Y$ and $\nu=\mu\circ \phi^{-1}$.
Here I take $X=[0,1), Y=[0,1)^2$ and $X'=[0,1)\cap \mathbb Q^c$, $Y'=([0,1)\cap\mathbb Q^c)^2$ then each has Lebesgue measure $1$. I chose irrationals because they have unique representation.
I define $\phi:X'\to Y'$ as $\phi(0.a_1a_2a_3a_4...)=(0.a_1a_3a_5...,0.a_2a_4a_6...)$ basically taking the odd and even parts in the decimal expansion. Then I saw that $\phi$ is bijective (injectivity follows due to unique representation in base 10, for irrationals).
I am not sure but I think this map is measurable and its inverse also is measurable. But I don't think the last property is true i.e. the value of the measures being same are not true in general.
Can you please help me to find a map, then, probably using mine or otherwise?
Any two uncountable Polish spaces $X$ and $Y$ (such as $\mathbb T$ and $\mathbb T^2$) are isomorphic via a Borel map with Borel inverse (without using this result I would say that it is unlikely to succeed). So you can push the second space $\mathbb T^2$ to the first, of course with an induced measure $\mu$ that may not be the Lebesgue measure. After that define a map $f(x)=\mu([0,x])$ on $\mathbb T$. This map (which is also an isomorphism) induces a new measure which is precisely the Lebesgue measure on $\mathbb T$, and so you only need to compose the two maps.