Cardinality and Measurability We can show that $\mathbb{R}$ and $\mathbb{R}^2$ or ($\mathbb{R}^n$) have same cardinality using the following one-to-one and onto mapping:
Say x = (0.123456789....)
Then f(x) = {(0.13579...),(0.2468..)}
My question is can we claim that the Borel sigma algebra in $\mathbb{R}$ has a correspondent sigma algebra (if it is, is it THE Borel sigma algebra in $\mathbb{R}^2$?) by mapping each Borel set D to $\mathbb{R}^2$ by f(D)? Why or why not could you give the argument??
Similarly (I am more certain of this)
When we have N^n and N using this mapping could we claim that the sigma algebra in N has a correspondent sigma algebra in N^n? Does the statement hold for N^infinity and N as well, if N^infinity and N have the same cardinality ( i am not sure about my last statement)? 
Thank you!
 A: The Borel sigma-algebra of $\mathbb{R}^2$ is defined as that generated by the subsets of $\mathbb{R}^2$ that are open in the product topology.  It is almost, but not quite, correct to say that this is equal to the collection of pointwise images of Borel subsets of $\mathbb{R}$ under your a correspondence given by splitting and interleaving decimal expansions.
The problem, as Ittay pointed out in a comment, is that this correspondence is not quite a bijection between $\mathbb{R}$ and $\mathbb{R}^2$.  Nevertheless, by changing the argument a bit, one can still show that $\mathbb{R}$ and $\mathbb{R}^2$ are Borel-isomorphic.  One way to do this is to get a Borel injection from $\mathbb{R}^2$ to $\mathbb{R}$ and then use the Borel version of the Cantor–Bernstein–Schroeder Theorem.
Indeed, a Borel injection $f: \mathbb{R}^2 \to \mathbb{R}$ can be defined as follows: Given $(x,y) \in \mathbb{R}^2$, let $x_0 x_1 x_2 \ldots$ and $y_0y_1y_2 \ldots$ be the unique decimal expansions of $x$ and $y$ respectively with the property that the digits are not eventually all 9's from some point on.
Then let $f(x,y)$ be the real number whose decimal expansion is $x_0y_0x_1y_1 \ldots.$  (For simplicity of notation I am ignoring the issue of where the decimal point goes, but this is easy to deal with.)
Unfortunately the range of this function is missing uncountably many real numbers (such as those whose even digits are eventually all 9's and whose odd digits are not) and the easiest way that I see to deal with this issue is to use the Cantor–Bernstein–Schroeder Theorem, although it may be overkill.
The sets $\mathbb{R}$ and $\mathbb{R}^2$ with these Borel structures (and indeed $\mathbb{R}^n$ for any positive integer $n$) are examples of 
standard Borel spaces, which are the objects one gets from Polish topological spaces by forgetting the topology and remembering only the Borel structure. The Borel isomorphism we established above proves a special case of a theorem of Kuratowski, which says that every uncountable Polish space is Borel-isomorphic to $\mathbb{R}$, so up to isomorphism there is only one uncountable standard Borel space.
The sets $\mathbb{N}$ and $\mathbb{N}^n$ with the discrete Borel structure (every subset is a union of countable many singletons and is therefore Borel in the usual topology) are Borel-isomorphic to one another.  They are examples of the countably infinite standard Borel space, which is also unique up to isomorphism.
The set $\mathbb{N}^\infty$ (also known as $\mathbb{N}^\mathbb{N}$) of integer sequences with the Borel algebra generated by the product topology is an uncountable standard Borel space—it is Borel-isomorphic to $\mathbb{R}$ rather than to $\mathbb{N}$.
A: The fact that there are $1-1$ mapping between two sets says almost nothing about the measure of the two sets. Consider a silly example of the Cantor set, you can show relatively easily that it has cardinality $c$ while having zero measure. 
The $f$ you defined from $\mathbb{R}^{1}\rightarrow \mathbb{R}^{2}$ is measurable (because it is continuous), and hence $f(D)$ does generate a $\sigma$-algebra in $\mathbb{D}^{2}$. But this $\sigma$-algebra is "twisted" in some sense. For example, the image of $(0.4,0.5)$ in the plane includes points "outside" of $(0.4,0),(0.5,0)$ like $(0.4,0.9)$ from $0.49$ or $(0.49,0.95)$ from $(0.4995)$. It is intuitively not the right choice for us to define a $\sigma$-algebra on the plane, where the open sets arise from open balls or open retangles. 
