Measure the "actual" number of dimentions in a multivariate distribution Consider a 3D multivariate normal distribution $x\sim N(0,\Sigma)$ where
$$\Sigma=\begin{bmatrix}1 &1 &0 \\ 1&1&0 \\ 0 &0& 1 \end{bmatrix}$$
Since $x_1$ and $x_2$ are fully correlated, one could say that this is actually a "2 dimensions" distribution.
We can also imagine an intermediate case with:
$$\Sigma=\begin{bmatrix}1 & 0.8 &0 \\ 0.8 &1 & 0 \\ 0 &0  & 1 \end{bmatrix}$$
that could actually be a "2.2 dimentions" distribution.
Is there a mathematical tool to represent this idea?

If you have a theoretical covariance matrix $\Sigma$, the rank tells you the number of dimensions. Whether you consider some of these dimensions insignificant from a substantive point of view is a substantive, not a statistical question.
If you have an empirical rather than theoretical covariance matrix $\hat\Sigma$, you could apply exploratory factor analysis (EFA). It is a technique for determining the underlying dimension of a group of variables each of which is supposedly generated as a linear combination of a smaller number of latent common factors plus an idiosyncratic factor (an error term, if you will). The way the number of dimensions is determined in EFA is a bit subjective, though.
