# Multiple integrals with Lebesgue theory?

From two measurable spaces $(\Omega_1, \mathcal A, \mu)$ and $(\Omega_2, \mathcal B, \nu)$, we can define another measurable space denoted $(\Omega_1\times \Omega_2, \mathcal A\otimes \mathcal B, \mu\otimes \nu)$ in wich we can establish double integrals and related theorems (as Fubini-Tonelli & Fubini-Lebesgue).

My question is : how to extend "product measure space" for $n$-dimensional spaces $\mathbb R^n$ defined as $\mathbb R^{\{1,..., n\}}$ ? Does an isomorphism preserve "measurability" of subsets ?

For example, suppose that $\mathbb R^2\cong \mathbb R\times \mathbb R$ through cannonical injection $\phi : (x_i)_{i\in \{1,2\}} \mapsto (x_1,x_2)$.
If these sets are equiped of the associated borelian $\sigma$-algebra, I think that $\phi$ is measurable... (sorry if I'm wrong, because I'm currently overviewing my courses on Measure and Integration).

If $\phi$ is actually measurable, what the measures on $\mathbb R^2$ and $\mathbb R \times \mathbb R$ have to satisfy to get :

$$\int_{\mathbb R^2} f\,\mathrm d\mu_{\mathbb R^2} = \int_{\mathbb R \times \mathbb R} f\circ \phi^{-1} \;\mathrm d\mu_{\mathbb R \times \mathbb R} "$$ I don't know if that really makes sense :/ But... I don't agree the following definition of $\mathbb R^n$ as $\mathbb R \times \mathbb R \times \cdots\times \mathbb R$ ($n$ times), because set theory does not allow to write this... moreover I am not fond of "recursive definition" of $\mathbb R^n$ (like that : $\mathbb R^n = \mathbb R^{n-1} \times \mathbb R$), and consequently, of product measure space on $n$-dimensionals :(

We can define $\mathbb{R}^n$ to be the set of all functions $n\to \mathbb{R}$, where $n=\{0,\dots,n-1\}$. We can also define, recursively, another family of sets by: $$\begin{cases} E_1:=\mathbb{R}\\ E_{n+1}:=E_n\times \mathbb{R}\end{cases}$$ We can show (by induction), that $E_n$ and $\mathbb{R}^n$ are homeomorphic, and so in particular isomorphic as measurable spaces with the Borel $\sigma$-algebra associated to the product topology.
We can also recursively define $n$-dimensional Lebesgue $m_n$ on $E_n$ measure by $m_{n+1}:=m_n\otimes m$, where $m$ is one-dimensional Lebesgue measure. By the remarks, this defines a regular Borel measure on $\mathbb{R}^n$ via the natural isomorphism, which then becomes a measure preserving isomorphism (measurable with measurable inverse).
We can also show $\mathbb{R}^n \times \mathbb{R}^k \cong \mathbb{R}^{n+k}$, which then proves $E_n\times E_k\cong E_{n+k}$ (this can also be showed directly by induction, keeping one index fixed).
If you want a more direct approach, you could read up on the construction of Haar measure on arbritrary locally-compact Hausdorff group, which $\mathbb{R}^n$ certainly is. This is, however, a tad more advanced, and you're likely to run into the Riesz representation theorem (which also allows for a direct construction, look at Rudin's Real and Complex Analysis, chapter 2).