As I was watching a seminar given by Alain Connes at IHÉS, I came to realize I had no idea on how to properly define what a real variable is. As far as I can remember, I've never been taught that, and I can't seem to find any answer on the Web.
According to Connes, a real variable is a map $f : X \rightarrow \mathbb{R}$ where $X$ is a set.
Now, Connes believes this widely-accepted definition poses a problem as it comes to the coexistence of continuous and discrete real variable, which I quite frankly find to be even more puzzling.
Therefore, I reckon one has the following definitions:
- a real variable is continuous if $f(X)$ has at least the cardinality of the continuum.
- a real variable is discrete if $f(X)$ is at most countable.
Connes' argument is the following: given a set $X$, if $f:X \rightarrow \mathbb{R}$ is a continuous real variable, then $\# X \geq \# \mathbb{R}$. Therefore if $g:X \rightarrow \mathbb{R}$ is another real variable, as $X$ has at least the cardinality of the continuum, which forces $g$ to be a continuous real variable, making it impossible to have coexistence of continuous and discrete real variable.
Therefore, my previous two definitions are misleading. Because, $g$ might actually take one value (send every element of $X$ onto $0 \in \mathbb{R}$ for instance), making it a discrete variable according to my definition.
Hence, how to correctly define what a real variable is? I can't seem to fathom Connes' definition. Once defined, how would you go at defining the continuous and discrete real variable?